Blue tits can do it: climbing and clinging on various structures, also upside-down, seems to be easy for this kind of birds.
https://ww2.rspb.org.uk/community/wildlife/homesforwildlife/m/hfwwildlife/35888.aspx
Also I have seen sparrows that could climb/cling (also upside-down) for a short moment on something. However, I have never seen that a collared dove "rock climbed" on something. Is this because a collared dove has more weight than a blue tit or have blue tits special forms of the feet?
Could it be possible that someone who owns a collared dove can train it to climb upside-down (rock climbers say sometimes "climbing on a overhang or roof")?