Spy-Versus-Spy

If you are looking for a cartoon, see Spy vs. Spy

Perhaps inspired by the use of SPECTRE as the villain in the James Bond movies rather than the communist villains of the books, a popular trope in spy films is to have two opposing organizations of agents who have a tenuous connection to a government if any. Typically, this is played for humor, and oddly, while some of the evil organizations are supposed to have quite horrible ideologies, their membership is generally depicted as being of a Punch Clock Villain nature.

Anime and Manga

 * Hellsing has a serious examples of various Church Militant groups opposed to one another.
 * Also serious: Darker than Black has constant infighting between intelligence agencies of all kinds, mostly to find out more about contractors and the Gate.

Comics

 * The Trope Namer is the famous Mad Magazine comic Spy vs. Spy which featured black and white trenchcoat-clad agents always trying to kill one another for no apparent reason. Occasionally a third spy, Lady Grey, would make an appearance and foil the both of them.
 * S.H.I.E.L.D. vs HYDRA.
 * W.A.S.T.E. vs E.M.P.I.R.E. vs M.O.T.T vs X.S.M. all to to be found in the pages of Matt Fraction's Casanova.

Film

 * Clearly inspired by Get Smart (below), the modern Blaxploitation parody Undercover Brother has the black organization BROTHERHOOD facing "The Man".
 * Combined with Murder, Inc. in Mrs. and Mrs. Smith which features opposing groups of assassins.
 * Our Man Flint and sequel had Z.O.W.I.E. vs. Galaxy (a parody of SPECTRE).

Literature

 * In a very strange way, this is how angels and demons tend to operate in Good Omens, as personified by Friendly Enemies Crowley and Aziraphale. Their situation is explicitly compared to two spies in the middle of nowhere who've been essentially forgotten by their respective governments.
 * Happens in some Tom Clancy novels, most notably The Cardinal of the Kremlin.

Live-Action TV

 * One of the most influential versions was that in Get Smart where the good guys CONTROL faced off against KAOS.
 * The Man from U.N.C.L.E. had U.N.C.L.E. vs. THRUSH.
 * Alias had several, all non-humorous. First was CIA versus SD-6, then CIA versus Covenant, then CIA versus whatever Irina Derevko's network was. Plus there were other spy groups, like K Directorate, that worked against the CIA and SD-6.
 * Chuck has a lot of this as well: CIA vs Fulcrum, CIA vs The Ring, CIA vs Volkoff Industries...

Video Games

 * Team Fortress 2 uses this basic framing device (with the heroically-evil RED and evilly-heroic BLU) to provide the deliberately flimsy rationale for the game.
 * Subverted in the fact that despite having all the hallmarks of the genre, RED and BLU are construction companies
 * Additionally, as Spies are one of the selectable classes, it literally becomes Spy vs. Spy in some cases.
 * The ACME Detective Agency and V.I.L.E. in the Carmen Sandiego games. ACME is basically a Brand X version of Interpol and V.I.L.E. pretty much just exists to be evil. Or to feed Carmen's ego, Depending on the Writer. They aren't exactly spy organizations, but the series borrows tropes from the spy genre enough that it really doesn't matter.
 * Ace Attorney Investigations is basically built around Interpol (represented by Lang and  vs. the smuggling ring, although Edgeworth seems to be the only one who does any actual detectiving.

Western Animation

 * S.H.U.S.H. vs F.O.W.L. in Darkwing Duck.
 * G.I. Joe and Cobra had elements of this, although the spy aspect of the series often took a backseat to the action.