Air carrier pilot reported during the climb they experienced severe turbulence resulting in significant excursions of speed; altitude and pitch.

Narrative

We departed ZZZ1 bound for ZZZ and I was PF. I expected turbulence for most of the flight so the flight attendants had been instructed to stay seated the entire flight. During the climb to FL360 in light turbulence; ATC reported that the ride was mostly smooth at that altitude and our performance ceiling was FL380; so I elected to continue the climb in the attempt to find smooth air. Climbing through FL350 we began to get some moderate chop and some St. Elmos fire on the windshield. As we continued to climb through about FL355; the turbulence began to rapidly intensify and I asked the FO (First Officer) to request a descent to FL340. Before he could do so there were multiple significant excursions of both speed and altitude trends as the turbulence increased to severe; the PLI appeared; and the aircraft rolled to the right. As I reached for the autopilot disconnect; the aircraft rolled sharply to the left; the stick shaker activated and the autopilot disconnected. I lowered the nose; regained airspeed and slowly brought the wings level as speed increased. I then ordered the FO to report the severe turbulence and our descent to FL340 as we descended back through FL350. The turbulence quickly abated and we entered smooth air. Once I got the aircraft stabilized; I re-engaged the autopilot; then called to check on the flight attendants. They reported no damage or injuries; and after a quick survey; I determined the aircraft was operating normally as well. We elected to continue the flight normally after notifying Dispatch of the severe turbulence encounter; and had a normal approach and landing in ZZZ. After deplaning was complete; I checked in with the flight attendants and did a quick debrief to make sure they were OK and that no passengers had made any significant remarks or reported any damage or injuries during the remainder of the flight. The aircraft was then written up for a severe turbulence encounter and was inspected overnight.