A CPR LJ35A FLC WAS CLRED FOR TKOF; BUT AFTER LIFTOFF THE ATCT CTLR DENIED THAT THEY WERE CLRED.

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I RECALL THE CTLR TELLING THE HELI TO REMAIN CLR OF RWY 4L&R. WE BOTH RECALL THE TWR THEN TELLING US THAT WE WERE CLRED FOR TKOF; FULL LENGTH RWY 4L; WHICH NECESSITATED A BACK TAXI OF SEVERAL HUNDRED FT. WE BEGAN THE BACK TAXI; AND I WAS UNSURE OF THE CLRNC ENOUGH THAT I ASKED THE PNF IF HE UNDERSTOOD AS I DID; THAT WE WERE CLRED FOR TKOF ON REACHING THE END. HE REPLIED THAT YES; THAT WAS CORRECT. I THEN ASKED HIM TO VERIFY THAT WITH THE TWR. HE CALLED AND SAID 'TWR; CONFIRM LEAR IS CLRED FOR TKOF.' TWR CAME RIGHT BACK WITH '...IS CLRED FOR TKOF FULL LENGTH RWY 4L.' BY THAT TIME; I WAS TURNING AROUND AT THE END; SO I KEPT IT ROLLING AND STARTED THE TKOF. DURING THE TKOF ROLL WE HAD THE HELI IN SIGHT AND IT QUICKLY BECAME APPARENT THAT IT WAS ANGLING ACROSS ONE RWY ABOUT 1/2 WAY DOWNFIELD. THE TWR SAW IT ALSO; AND CALLED THE CAPT AND TOLD HIM TO MAKE AN IMMEDIATE R TURN. HE DID SO; AND DID NOT CROSS OVER US. JUST AS WE WERE AIRBORNE; TWR CALLED AND SAID SOMETHING ABOUT TKOF THAT I DID NOT FULLY UNDERSTAND; BUT I SURMISED THAT THERE WAS SOME CONFUSION. I CALLED THE TWR AND SAID 'SIR; WE VERIFIED TWICE THAT WE WERE CLRED FOR TKOF.' THE CTLR THEN SAID TO FLY THE DEP AND CONTACT DEP. AFTER CONTACTING DEP; THE CTLR PASSED ALONG A REQUEST THAT WE CALL MIDWAY TWR VIA PHONE. UPON REACHING HOME BASE; WE DID SO AND TALKED TO A CTLR THAT IDENTED HIMSELF AS THE SUPVR. HE SAID HE WAS STANDING BEHIND THE CTLR TALKING TO US DURING THE TKOF; BUT COULD NOT HEAR WHAT WAS SAID AND WANTED OUR COMMENTS ON WHAT HAPPENED. I ASKED HIM IF HE HAD LISTENED TO THE TAPE AND HE SAID NO. I THEN RELATED THE SEQUENCE JUST AS I DID HERE. HE SAID HE THOUGHT WE MAY HAVE HEARD THE TKOF CLRNC GIVEN THE HELI AND CONFUSED IT WITH OURS. I REPLIED THAT I DID NOT THINK THAT WAS POSSIBLE; SINCE WE CALLED BACK TO VERIFY THE CLRNC. BOTH PLTS (WE WERE BOTH ON THE LINE) THEN SAID WE WOULD CERTAINLY LIKE TO HEAR THE TAPE TO VERIFY WHAT WAS SAID. HE THEN SAID THAT IF HE WERE TO PULL THE TAPES; HE WOULD BE OBLIGATED TO FILE A RPT AND POSSIBLY GO FURTHER; AND HE WOULD LIKE TO JUST TALK TO US AND STOP AT THAT. WE BOTH SAID WE HAD NO PROB WITH THAT. I REALLY BELIEVE WE WERE CORRECT IN THIS; THAT IT WAS A MISTAKE IN THE TWR. BUT I WOULD JUST AS SOON NOT GO FURTHER. WE DISCUSSED HOW TO PREVENT A FUTURE SIMILAR PROB BY BEING MORE VIGILANT ON CLRNCS AND CONFIRMING AND VERIFYING EVEN MORE TO ENSURE EVERYONE IS IN THE SAME LOOP.