
559 F.3d 1041 (2009)
INTERNATIONAL UNION OF OPERATING ENGINEERS, Plaintiff-Appellee,
v.
COUNTY OF PLUMAS, Defendant-Appellant.
No. 07-16001.
United States Court of Appeals, Ninth Circuit.
Argued and Submitted December 10, 2008.
Filed March 20, 2009.
*1042 Michael E. Chase, Boutin Dentino Gibson Di Giusto Hodell Inc., for the appellant.
Steven W. Welty and James R. Traber, Mastagni, Holstedt, Amick, Miller, Johnsen & Uhrhammer, for the appellee.
Before: SIDNEY R. THOMAS and RICHARD A. PAEZ, Circuit Judges, and VAUGHN R. WALKER,[*] Chief District Judge.
THOMAS, Circuit Judge:
James M. Cain, whose novels were often adapted into film noir, described his body of work by saying "I write of the wish that comes truefor some reason, a terrifying concept." The plight of Plumas County would have doubtless piqued his interest.
Plumas County thought that federal court would provide a more hospitable forum for its defense against a suit seeking to compel arbitration, so it removed the action to federal court. The district court ordered Plumas County to arbitrate. Now, the County argues that the district court had no business deciding the question *1043 because the court lacked subject matter jurisdiction over the case that the County removed. We conclude that Plumas was not barred from raising the jurisdictional argument and that the district court lacked subject matter jurisdiction. We remand to the district court the question of whether attorneys fees should be assessed.

I
The County of Plumas laid off five employees, allegedly for budgetary reasons. The International Union of Operating Engineers filed a grievance against Plumas, claiming the lay-offs were pretext for disciplinary terminations. The Union sought to arbitrate the grievance under its collective bargaining agreement with the County. The County asserted that the layoffs were not subject to arbitration.
The Union then filed a petition in the California Superior Court to compel arbitration. The Union contended that the County had violated its published personnel rules governing disciplinary actions and layoffs. The Union averred that federal law controlled construction of the collective bargaining agreement but that state courts had concurrent jurisdiction.
The Union argued that the National Labor Relations Act, 29 U.S.C. § 152, as well as a California statute enforcing arbitration clauses, Cal.Civ.Proc.Code § 1281, compelled arbitration.
The County then filed a notice of removal of the action pursuant to 28 U.S.C. § 1331, claiming that the Union's "right to relief on its claims depends on this Court's resolution of substantial federal questions." The Union did not move to remand the case back to state court.
The federal district court granted the Union's motion to compel arbitration. It held that the collective bargaining agreement's binding arbitration provision created a presumption of arbitrability. This presumption required the County to establish "positive assurance" that the arbitration clause was not susceptible to an interpretation that would cover the dispute. The court concluded that the County had not sufficiently addressed the Union's contention that the layoffs were pretext for disciplinary action and compelled binding arbitration. Additionally, the court ordered the County to enter advisory arbitration to resolve the Union's claim that the County improperly invoked its layoff powers.
On appeal, the County reverses course and argues that the district court lacked subject matter jurisdiction. The County asks us to vacate the order compelling arbitration and remand the case to state court. The Union chose a state forum in the first instance. However, having prevailed before the district court, it now endorses the existence of federal subject matter jurisdiction with a bit more enthusiasm.

II
Although the County originally asserted the existence of federal subject matter jurisdiction when it removed this case to federal court, we conclude that nothing precludes it from challenging subject matter jurisdiction on appeal. The general rule, of course, is that "absent any good explanation, a party should not be allowed to gain an advantage by litigation on one theory, and then seek an inconsistent advantage by pursuing an incompatible theory." New Hampshire v. Maine, 532 U.S. 742, 749, 121 S.Ct. 1808, 149 L.Ed.2d 968 (2001) (quoting 18 Charles Alan Wright, Arthur R. Miller, & Edward H. Cooper, Federal Practice and Procedure § 4477 (1981)). Nonetheless, "[a] party may raise jurisdictional challenges at any time during the proceedings." Attorneys Trust v. Videotape Computer Prods., 93 F.3d 593, 595 (9th Cir.1996) (quoting *1044 May Dep't Store v. Graphic Process Co., 637 F.2d 1211, 1216 (9th Cir.1980)).
Thus, even though the County asserted subject matter jurisdiction in its removal notice, it is not precluded from challenging subject matter jurisdiction on appeal. See United States v. Ceja-Prado, 333 F.3d 1046, 1050 (9th Cir.2003) (noting "the general principle that jurisdictional concerns trump equitable considerations"); Attorneys Trust, 93 F.3d at 594-95 (allowing a "disappointed plaintiff" who originally brought suit in federal court to challenge federal jurisdiction on appeal). Put another way, the County may be guilty of chutzpah, but we must consider the merits of its argument anyway. United States v. Sar-Avi, 255 F.3d 1163, 1166-67 (9th Cir. 2001).

III
We conclude that the district court lacked subject matter jurisdiction over the Union's petition. The order compelling arbitration must therefore be vacated and the action returned to state court.

A
The National Labor Management Relations Act ("LMRA"), 29 U.S.C. § 152, does not create federal jurisdiction in this case, as the County initially contended and the Union now contends. The LMRA does not govern the collective bargaining agreement at issue because the County is not an "employer" within the aegis of the LMRA. The LMRA provides that "[t]he term `employer' includes any person acting as an agent of an employer, directly or indirectly, but shall not include the United States... or any State or political subdivision thereof." 29 U.S.C. § 152(2) (emphasis added). Because the County is a political subdivision of the State, see NLRB v. Natural Gas Util. Dist., 402 U.S. 600, 605-09, 91 S.Ct. 1746, 29 L.Ed.2d 206 (1971) (defining a political subdivision), it does not fall within the reach of statute. Ayres v. Int'l Bhd. of Elec. Workers, 666 F.2d 441, 444 (9th Cir.1982) ("[S]ection 301(a) of the Act, 29 U.S.C. § 185(a), does not grant this court jurisdiction over the claims of an individual employed by a political subdivision of a state.").

B
Nor does federal jurisdiction arise in this case because California has adopted principles of federal labor law. Of course, we recognize that "a case may arise under federal law `where the vindication of a right under state law necessarily turn[s] on some construction of federal law.'" Merrell Dow Pharm., Inc. v. Thompson, 478 U.S. 804, 808, 106 S.Ct. 3229, 92 L.Ed.2d 650 (1986) (quoting Franchise Tax Bd. v. Constr. Laborers Vacation Trust, 463 U.S. 1, 9, 103 S.Ct. 2841, 77 L.Ed.2d 420 (1983)).
As the Supreme Court has explained:
How and when a case arises under the Constitution or laws of the United States has been much considered in the books. Some tests are well established. To bring a case within the statute, a right or immunity created by the Constitution or laws of the United States must be an element, and an essential one, of the plaintiff's cause of action. The right or immunity must be such that it will be supported if the Constitution or laws of the United States are given one construction or effect, and defeated if they receive another. A genuine and present controversy, not merely a possible or conjectural one, must exist with reference thereto, and the controversy must be disclosed upon the face of the complaint, unaided by the answer or by the petition for removal. Indeed, the complaint itself will not avail as a basis of jurisdiction in so far as it goes beyond a *1045 statement of the plaintiff's cause of action and anticipates or replies to a probable defense.
Gully v. First Nat'l Bank, 299 U.S. 109, 112-13, 57 S.Ct. 96, 81 L.Ed. 70 (1936) (citations and internal quotation marks omitted) (emphasis added).
"Arising under" federal jurisdiction only arises, then, when the federal law does more than just shape a court's interpretation of state law; the federal law must be at issue. Here, the state law right to compel arbitration does not turn on a construction of federal law. Rather, it is influenced by an application of the federal law to the arbitration clause. California courts are not bound by federal labor law, they have merely voluntarily adopted parts of it. See, e.g., United Transp. Union v. S. Cal. Rapid Transit Dist., 7 Cal.App.4th 804, 9 Cal.Rptr.2d 702, 707 (1992) ("[W]e hold that we are not bound by the definition of `employer' in the Labor-Management Relations Act when we determine whether the dispute between the Union and the District should be arbitrated.").
In addition, when "an alternative theory of relief exists for each claim alleged in the complaint, one not dependent upon federal law," federal question jurisdiction is defeated. Ultramar America, Ltd. v. Dwelle, 900 F.2d 1412, 1414 (9th Cir.1990) (citing Christianson v. Colt Indus. Operating Corp., 486 U.S. 800, 810, 108 S.Ct. 2166, 100 L.Ed.2d 811 (1988)). The Union's petition includes alternative theories for its claims founded solely on state law.
For these reasons, under the facts presented by this case, we must reject the Union's argument that federal subject matter jurisdiction existed because its California state law claims arose under federal law.

C
Because the Union's petition to compel arbitration was not governed by the LMRA and its claims did not "arise under" the Constitution or federal law within the meaning of Merrell Dow Pharmaceuticals, we conclude that the district court lacked subject matter jurisdiction.

IV
In the archetypical James Cain novel, two parties wish an outcome and both endeavor to bring it to fruitionsometimes together and sometimes independentlyusually employing less than savory means.[1] Here, both the County and the Union have asserted the existence of federal jurisdiction at various times when it suited their interests.
Because the County initiated this foray into federal court, the Union requests that if we conclude federal jurisdiction is absent, we should award the Union its fees. We decline to reach that question and instead remand it to the district court. We do so with some hesitation because the Eastern District of California has one of the most pressing dockets in the nation. However, bearing in mind the admonition that one ought to temper the rule of law with poetic justice, we conclude that the district judge is in the best position to decide which party wasted the court's time, and whether fees or other sanctions should be imposed.
We reverse the judgment of the district court and remand the case with instructions first to determine whether fees, costs, or sanctions should be imposed and *1046 then to remand the case to the California Superior Court in and for the County of Plumas.
REVERSED AND REMANDED WITH INSTRUCTIONS.
NOTES
[*]  The Honorable Vaughn R. Walker, Chief District Judge for the Northern District of California, sitting by designation.
[1]  See, e.g., JAMES M. CAIN, THE POSTMAN ALWAYS RINGS TWICE, DOUBLE INDEMNITY, MILDRED PIERCE, AND SELECTED STORIES (Every man's Library Classics 2003).
