AN ATX C650 DOES NOT MAKE THE R TURNOUT TOWARDS PSP VOR ON A NON RADAR DEP PROC FROM PSP.

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WE WERE GIVEN A CLRNC TO FLY A PLT NAV-SID FROM PALM SPRINGS; CA; THE CATHEDRAL SEVEN DEP. RADAR SVC WAS OUT DURING OUR DEP TIME. WE DID NOT FLY THE SID AS PUBLISHED. THE CAPT AND I HAVE FLOWN OUT OF PALM SPRINGS NUMEROUS TIMES USING THE PSP 3 DEP WHICH IS A RADAR VECTOR SID. DURING PAST DEPS FROM PSP I CANNOT REMEMBER FLYING THE CATHEDRAL SEVEN SID BEFORE. UPON ROTATING OUT OF PSP WE WERE TOLD TO CONTACT PSP DEP. I CONTACTED PSP DEP AND TOLD THEM WE WERE OUT OF 1000 FT FOR 9000 FT RWY HEADING. THE CTLR TOLD US TO CLB AND MAINTAIN 7000 FT. WE WERE THEN CLRED TO 9000 FT AND TOLD TO RPT OUT OF 7000 FT. THE FREQ WAS EXTREMELY BUSY AND WE WERE UNABLE TO CONTACT DEP UNTIL REACHING 8000 FT. WE WERE THEN TOLD TO CONTACT ZLA. UPON CONTACTING ZLA I RPTED OUT OF 8300 FT FOR 9000 FT HDG 310 DEGS. THE CTLR STATED MAINTAIN 9000 FT. I REQUESTED A HIGHER ALT AND WAS TOLD TO CONTACT ZLA ON 126.35. I CONTACTED ZLA AND TOLD THEM WE WERE OUT OF 8800 FT FOR 9000 FT. THE CTLR STATED; 'TURN R TO A HDG OF 340 DEGS AND CLB AND MAINTAIN 13000 FT.' AT THIS TIME THE CTLR WORKING THE FREQ 126.35 ASKED US WHAT OUR CLRNC WAS. WE STATED WE WERE CLRED VIA THE CATHEDRAL SEVEN DEP; BUT THOUGHT WE WERE TOLD TO FLY RWY HEADING INITIALLY. ZLA TURNED US TO A 340 DEG HDG AND TOLD US TO CLB TO 13000 FT AND THEN A SHORT TIME LATER CLRED US TO FL230 DIRECT PALMDALE. I THOUGHT WE WERE TOLD TO FLY RWY HEADING INITIALLY EVEN THOUGH THE PLT SID STATES TURN R TO THE PSP VOR. I HAVE FLOWN OUT OF PSP MANY TIMES AND ALMOST ALWAYS HAVE BEEN VECTORED TO PSP VOR OR A VECTOR HEADING TO JOIN V-386 TO PMD. I WAS EXPECTING A RADAR VECTOR. OUR INACTION WAS THAT WE BRIEFED THE SID AND DIDN'T FLY IT BECAUSE WE THOUGHT WE WERE GOING TO GET A TURN TOWARDS THE PSP VOR. AFTER THE INCIDENT THE CAPT AND I DISCUSSED THE EVENT AND AREAS OF ENHANCING PRE-DEP BRIEFINGS AND BETTER CRM.