I myself have never had a testicular exam.
If I do, there are two possible scenarios:
Female doctor- then I would not touch/ lift the penis myself, because i would want her to touch me as much as possible, especially in the sensitive genital area.
I would want her to lift or move around my penis as needed to get access to testes/ scrotum.
I would however stand with legs slightly apart and bend down so she gets full view and entire access.
If foreskin retraction is required, I would want her to handle that as well. I hope full retraction would not be needed because I have tight foreskin and if dry and harsh it is painful to retract abruptly beyond the ridge. So I hope partial retraction to examine the front of glans would suffice.
If she retracted me, I hope that she would also bring the skin back to cover the glans completely, because I'm most comfortable in this position.
If the doc is Male, I would lift up my penis myself, although still need to give him full access to scrotum. I would want to retract and bring back my foreskin myself, if needed for the exam. Obviously because I am straight I'm not happy with another guy touching me if it can be avoided.