Tidally locked, and yet spinning? Reading about Uranus almost 90° tilt, I was wondering if some rocky planet with mass concentration at one pole could possibly spin around its own axis, while still being locked to its star? Which means the more massive pole always points toward the star?
I think what you are envisioning is having one pole always pointed towards the primary star. So the planet would rotate about that pole, but then the direction of the pole would change over the course of a year to keep the pole pointing towards a star.

This phenomenon of the direction of the pole changing is called "precession". The Earth does it over a period of 26,000 years, but the pole only traces out a fairly small circle in the sky, rather than twisting about a perpendicular line like you envision.

I think your scenario is improbable at best. Planets are like gyroscopes and resist having their axes of rotation be twisted about a perpendicular axis like that. Even a planet with mass concentrations at both poles (maybe a long cigar shape) to maximize tidal force would probably end up losing the perpendicular rotation to tidal friction, and just end up rotating slowly about an axis nearly parallel to the axis of the orbit.