There were similar rivalries in Germany (e.g. Gestapo vs. SS vs. Wehrmacht) -- different state rackets competing with each other is a common feature of fascism.
The Allies were not immune to this either, especially the Soviets (NKVD vs military). In the democratic countries it was more on the individual level, with certain powerful people fighting to get their way (i.e. MacArthur).
From 1936 onwards, the Gestapo was entirely subordinate to the SS, as were most of the police organizations in Germany. There was no rivalry between it and the SS, nor with the Wehrmacht.
