Why do some transformers work just fine when powered from the secondary and other don't?

I was using a small (500VA) 240 to 120 step down transformer as a step up for a small load, and I could only get 200V when powering the 120V winding with 120V.

The turns ratio is calculated to account for the losses inherent in the transformation.
Instead of adding more power to compensate for the 240 to 120 losses...you may be deducting twice that value when going from 120 to 240.

The effect is much less noticeable in large transformers so you may not notice it in those cases.

Larry, are you saying that the unloaded voltage was only 200VAC? Did you verify it by driving the primary at 240VAC? I've driven quite a few backwards and everything followed the turns ratio.

I wonder if it was really 208-120.

It really is a 240 to 120 transformer. I checked it with 240 Input and I got 120 Out. I was told this specific transformer is a cheap design. Low power, 50/60 Hz operation.

I have been told that there is now a new(?) rating on other larger dry transformers stating that it is suitable for reverse operation.

I am no longer at that facility to get manufacturer and model numbers.

450.11(B) of the 2014 NEC has language about feeding the secondary of a transformer.

Winchester Speedpark is up there in NH. Nice place...had lots of fun there.

Just down the road a couple of miles. Some nights we can just barely hear the cars.

The transformer supply will be definitely damaged (when there is no protection scheme) due to nearly zero resistance of the primary winding. Moreover, it is obvious that you cannot see any voltage signal in the secondary! So take the advice of licensed electrical contractors and use transformers in your AC.

But the beauty of transformers is that they allow us to have more than just one winding in either the primary or secondary side. Transformers which have more than one winding are known commonly as Multiple Winding Transformers.
Note that transformers are only used for AC systems because of the phenomena of mutual induction between the coils. We recommend you review the functionality of a transformer again and get the gist of why we use transformers in AC power systems.

Transformers, on the other hand, are able to convert levels of AC voltage and current at very high efficiencies with no moving parts, making possible the widespread distribution and use of electric power we take for granted.
In all fairness, it should be noted that motor supply sets have not necessarily been obsoleted by transformers for all applications.