The Prisoner.
Of course, there were all those Gerry & Sylvia Anderson series - I remember Supercar, Fireball XL5, Stingray, Thunderbirds, Captain Scarlet, Joe 90, and Secret Service. Most, if not all, of them had different start and end music. Sometimes the difference was just having lyrics (also see Taggart), sometimes (eg Stingray) they were completely different tunes.
Thunderbirds - I was in the CCF at school, and the CO insisted on the Thunderbirds theme tune (which is a march) for our march-past. For some reason, that hasn't damaged me for life. Or perhaps it has.