I have sciatic back pain. How can I find out whether the pain caused by a torn rotor cuff surgery I recently had?

Not related. Sciatic nerve pain is caused by an irritation of the nerves that come from the low back. A rotator cuff issue will have no bearing on sciatica.
Simple - it isn't. It is most lijely unrelated unless you have modified how you were moving and lifting or being more sedentary as a result of the surgery. It is unlikely that it would be related to your positioning for the surgery. If you fell being transferred from the or table i would expect you would have been informed and you would have had other evidence of this unlikely scenario.