B757-200 FLT CREW BEGIN TO TAXI PRIOR TO RECEIVING A SALUTE.

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AFTER 2 NORMAL ENG START DURING PUSHBACK; CAPT SET PARKING BRAKE AND CLRED PUSH CREW TO DISCONNECT. THE DISCONNECT POINT WAS ONLY 200 FT FROM THE HOLD SHORT POS FOR THE RWY. I (THE FO) PREEMPTIVELY CALLED FOR TAXI CLRNC IN AN ATTEMPT TO ESTABLISH TKOF PRIORITY (PROVO WAS USING A SINGLE FREQ FOR GND/TWR/APCH TO CTL SUBSTANTIAL PVT AND COMMERCIAL TFC TO ONE RWY WHICH REQUIRED BACKTRACK FOR TKOF). PROVO FIRST SAID 'TAXI TO HOLDING POS FOR RWY 10 ON ALPHA;' THEN ASKED IF WE COULD ACCEPT 'BACKTRACK INTO POS AND HOLD.' I RELAYED THIS TO THE CAPT (HE WAS ON THE INTERPHONE ASKING THE #1 FLT ATTENDANT IF SHE WAS READY SINCE IT WAS GOING TO BE A VERY SHORT TAXI); HE REPLIED 'YES;' WHICH I RELAYED TO PROVO; WHO CLRED US TO DO SO. I VISUALLY SCANNED FORWARD RIGHT AND THE R WING AREA AND CALLED 'CLR RIGHT.' THE CAPT CALLED 'CLR LEFT;' RELEASED THE PARKING BRAKE AND VERY SLOWLY NUDGED THE PWR UP. JUST AS THE ACFT STARTED ROLLING FORWARD; THE CAPT ABRUPTLY STOPPED THE ACFT AS HE SAW THE TUG AND PUSH CREW BACKING OUT FROM BENEATH THE NOSE. THE ACFT HAD MOVED NO MORE THAN 1 FT. ABOUT 30 SECONDS HAD LAPSED SINCE THE PUSH CREW HAD BEEN CLRED TO DISCONNECT. LUCKILY; SO LUCKILY; NO ONE WAS HURT (THE TUG AND PUSH CREW MUST HAVE ALREADY STARTED TO BACK AWAY AS THE ACFT STARTED TO MOVE). THERE WAS NO EQUIP CONTACT OR DAMAGE. BUT THERE WAS A BUNCH OF SELF-RECRIMINATION ON THE FLT DECK AS TO HOW WE COULD HAVE BEEN SO COMPLACENT. SOME PROFOUND RELEARNING TOOK PLACE. I HAD FAILED TO CONFIRM WITH THE CAPT THAT HE'D SEEN THE 'PIN AND SALUTE' NOR DID I CONFIRM THE TUG AND PUSH CREW IN SIGHT AND WELL CLR; BEFORE CALLING 'CLR RIGHT.' NEVER AGAIN.