A320 Captain reported encountering wake turbulence in cruise at FL340 from an opposite direction air carrier aircraft equipped with winglets at FL350.

Narrative

Close to BDROC waypoint which is west of MTJ VOR; [we] encountered moderate wake turbulence off another aircraft flying above us. Prior to this encounter the ride was smooth. We were flying westbound at FL340. Denver Center advised of an aircraft flying at FL350 at our 1200 position. Both the FO (First Officer) and I saw the aircraft and commented that it looked lower than 1000 ft. separation. TCAS readout was 900 ft. above our altitude for the approaching aircraft. All three altimeters read 34;000 in our aircraft. The aircraft approached us head on to the point where we could not see the livery to determine what airline it was or what type of aircraft. Seeing winglets it was probably a B757 or A321NEO. We both noted the winds at FL340 were 239 at 28 kts. which was less than 10 degrees off our heading to the left. A few minutes later we encountered moderate wake turbulence and after about 15 seconds the aircraft began to roll left at a good rate. Throughout the turbulence until the left roll; the autopilot was on and maintaining altitude. When the aircraft started to roll left; I clicked off the autopilot and rolled the aircraft to the right. After approximately another minute we flew out of the wake turbulence. There were no deviations in altitude or heading. During this wake turbulence encounter ATC was calling for a frequency change. After flying the airplane; we responded; reported the wake turbulence encounter and switched frequencies. We both forgot to ask what type of aircraft flew over us. After another frequency change; another sector asked if the aircraft had any damage or if there were any injuries. We looked at all the systems pages on the ECAM and couldn't determine any damage. There were no passenger or Flight Attendant injuries. Thankfully; all the passengers just happened to be seated. The Flight Attendants took their jump seats when they felt the turbulence even before we could make a PA or call them. This isn't the first time I have encountered wake turbulence from another aircraft passing within 1000 ft. of our aircraft. This is the most severe encounter and the most roll I have encountered. We both did not say 'Upset'. But we were both guarding the controls and ready to take control if required. Thinking back on it; I think we didn't say the word upset because the aircraft wasn't in an upset state but I wonder if I would have said it if the situation got worse. We definitely had positive control of the aircraft and both pilots were monitoring the situation. I wonder if the FAA has done an evaluation on the new aircraft with large winglets coming into the market for wake turbulence probability. Especially with light winds aloft.