Is there a practical reason that the roles of the primary and secondary of a simple single-phase transformer cannot be swapped?

Say you have a transformer (from the old vacuum tube days) that has rated for primary 120 VAC 1 A and secondary 12.6 VAC 10 A (so there is about 0.120 kVA for this transformer).

Is there any reason that one cannot take that transformer and drive the secondary with 12.6 VAC and up to 10 A and get from the primary power at 120 VAC up to 1 A?

I cannot imagine a reason that, as long as the specs on the windings are kept that the roles of the primary and secondary windings of a simple power transformer cannot be swapped. But I would be happy to learn differently.
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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.
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.