MD80 CLBING OUT OF FL230 FOR FL270 EXPERIENCED CONFLICT WITH B737 OPPOSITE DIRECTION LEVEL AT FL240.

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WE DEPARTED PSP. WX WAS CLR WITH UNLIMITED VISIBILITY. I WAS FLYING THE ACFT; AND WE WERE CLRED TO FL230 BY ZLA. AS I WAS SMOOTHLY LEVELING OFF AT FL230; ATC ISSUED CLB CLRNC TO FL270. I ONCE AGAIN SMOOTHLY STARTED MY CLB ONLY TO RECEIVE A TA. I LOOKED AT THE TCAS; AND SAW TFC AT +900 FT JUST 2.0 NM AT THE 11:30 O'CLOCK POS. I LOOKED UP AND SAW A B737 RAPIDLY APCHING US IN MY WINDOW. AT THAT TIME; I RECEIVED AN RA TO REDUCE CLB. I SMOOTHLY LEVELED THE ACFT; AND THE B737 PASSED JUST 600-700 FT OVERHEAD. WE IMMEDIATELY ADVISED ATC OF THE RA; AND INFORMED HIM THAT WE WOULD BE FILING A RPT. HE RESPONDED THAT THE TFC WAS AT FL240; AND WAS BEHIND US AND THAT IT WAS NO FACTOR. I THEN JUMPED ON THE RADIO AND TOLD THE CTLR THAT HE HAD JUST CLRED US TO FL270 THROUGH THE B737'S ALT OF FL240. I DID NOT REMEMBER WHAT HIS RESPONSE TO THAT WAS. AT THE TIME OF THIS NEAR MISS; I WAS HAND FLYING THE ACFT; AND HAD IT BEEN ON AUTOPLT; I BELIEVE WE PROBABLY WOULD HAVE HAD A COLLISION WITH THE B737. THE AUTOPLT CLB WOULD HAVE BEEN FASTER; AND SINCE I HAD MY HAND ON THE YOKE; IT ENABLED ME TO RESPOND MUCH FASTER. ALSO; HAD IT NOT BEEN FOR THE TCAS; THERE IS A GOOD CHANCE THAT WE WOULD HAVE NOT SEEN THE B737 UNTIL IT WAS TOO LATE. THE TFC WAS COMING AT AN ANGLE AND I COULD NOT SEE HIM DUE TO THE WINDOW FRAME OBSTRUCTING MY VIEW. WHEN THE CTLR HANDED US OFF; HE SAID SORRY FOR THE CONFUSION. I DO NOT THINK HE REALIZES HOW CLOSE IT COULD HAVE BEEN.