Can roles of primary and secondary windings be reversed?
You can do exactly that.
Respect the maximum voltage on each winding. Transformers cannot be over-volted and still behave themselves.
Due to the various small non-idealities, winding resistance and leakage inductance being the major ones, you will notice that the voltage ratio one way when under load is slightly less than the reciprocal of the ratio the other way.
Yes, that'll work - with one caveat.
A carefully designed transformer will take into account the internal losses, to minimise them while reducing cost. So the primary may be wound with slightly thicker wire for lower resistance than in an "ideal" transformer, or the secondary with slightly thinner wire because it only has to handle the output power, minus the losses.
Driving a transformer backwards, these changes work against you, so I would de-rate it to 90% - or maybe 80% - of its rated power to be on the safe side - that is, 120V at 0.8 or 0.9A rather than 1A - and as Neil says you may also observe a minor discrepancy in the voltage ratios.
A simple answer is not always right. The first problem you have is your quoted ratings. 120 V @ 1 A - 120 VA. If the load is entirely resistive, ie, there is no reactive load, then the maximum you can get out of the secondary will be about 12.6 V @ 9 A, if the efficiency is 95% - not 10 A. Brian Drummond's answer is correct. I have designed transformers and been a production manager in a transformer manufacturing plant. I know what he and I are talking about - and he's correct. There will always be losses that the primary is designed to handle. So, if you reverse feed your transformer, the previously dedicated secondary will have to cover the magnetising, copper and eddy losses. So, you will get less power out of the 120 Vac winding. However, there is one more point: the primary winding will generally have a higher insulation from the core of the transformer, than the secondary. So, you need to consider carefully the source of the power you feed into the previous secondary.