To me, one can ("put on" or "wear") and "take off"all the mentioned items in the subject of this thread; I have made some examples. Please consider them and let me know if my sentences sound incorrect to you. Meanwhile, I think using either 'put on' or 'wear' in these senses do not make any semantic nuance in the sentences bellow; I would appreciate it if you specify the examples where using one of these two mentioned verbs convey a different connotation:
1) Hat / Cap and etc.
- I thought its cold outside and (put on / wore) my Knit cap; but the weather was good and I (took it off / took off it).
2) Tie and so on
Do you know that man who has (put a tie on / put on a tie / worn a tie)?
Because of an old habit I (take my tie off / take off my tie) at the time of dance.
It’s too sunny; I need to (put on / wear) my sunglasses.
Take of your glasses.
Most of the women love to put on / wear expensive jewels in ceremonies
Please take off your jewels.