C650 WAKE TURB ENCOUNTER WITH SAME ALT; OPPOSITE DIRECTION B757 DUE TO ATC OPDEV.

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WE WERE AT FL360 AND ASKED FOR A CLB TO FL380 BECAUSE WE WERE OUT-RUNNING A CITATION EXCEL AT FL380. THE ZDC CTLR CLBED US TO (OPPOSITE DIRECTION) FL370 AND MOMENTS LATER CHANGED US OVER TO THE NEXT WASHINGTON CTLR FREQ. AS WE ENTERED THE FREQ; WE HEARD THE CTLR TURNING AN ACR ACFT AND APOLOGIZING FOR 'FORGETTING ABOUT...(UNINTELLIGIBLE).' AT THAT TIME WE SAW THE TFC DIRECTLY IN FRONT OF US AND IN A VERY TIGHT TURN. I LOOKED DOWN AT OUR TA UNIT (NOT A REAL TCAS WITH LIMITED 6 MI RANGE); I COULD SEE THAT HE WAS ALSO AT FL370 (ZERO ALT DIFFERENCE). WE PASSED BEHIND HIM BY WHAT SEEMED TO BE LESS THAN 1 1/2 MI. SEEING THAT WE WERE NOT IN IMMEDIATE DANGER; I PUNCHED OFF THE AUTOPLT KNOWING THAT WE WERE GOING TO GO THROUGH HIS WAKE AND SINCE HE WAS IN A TURN THERE WASN'T ANY WAY FOR ME TO AVOID IT OTHER THAN TO CLB A FEW HUNDRED FT POSSIBLY CREATING YET ANOTHER UNSAFE CONDITION. I ELECTED TO STAY ON ALT. EVEN THOUGH THE TURB LASTED ONLY A FEW SECONDS; WHEN WE HIT HIS WAKE IT WAS LIKE HITTING A BRICK WALL. THE CTLR APOLOGIZED TO THE OTHER CREW AND THEN APOLOGIZED TO US AND CLRED US DIRECT TO ILM; EVEN THOUGH WE WERE PREVIOUSLY CLRED DIRECT TO ILM BY PREVIOUS CTLRS. THE FLT CONTINUED WITHOUT FURTHER INCIDENT. I BELIEVE THE CTLR ACCEPTED THE OPPOSITE DIRECTION TFC AND FORGOT ABOUT THE OTHER FLT. I BELIEVE THE CTLR GAVE HIMSELF A 'DEAL' AND YOU WOULD HAVE THOUGHT THE CTLR HANDING US OFF WOULD HAVE ALSO SEEN THE OTHER FLT AND WOULDN'T HAVE ASKED TO CLB US FROM FL360 TO FL370.