Engine mount, right driver side as suspected.

[Ghost-Flame]

Yup, sounds like a broken motor mount. Never had one in the X but had sveral in Fords and VW's I've owned and what you describe is exactly how those that failed on me acted. The quick jolt of rpm torques the engine away from the broken mount. I would replace both because of stress on the non broken mount.

This can be a DIY I believe you can use a jack to lift the engine to get at the motor mounts, but, I'm not sure. Also an engine lift will work if you have access to one.