Rewiring a small appliance
The transformer can be replaced, 120 /110v transformers are the same thing just different dates when they were manufactured. For many years we called line voltage 110, later 115 now 120 so they by definition are the same to the national electric code in the US. The important part is the output voltage and the size or how many amps / watts it is rated for. The new transformer needs to be rated the same or larger for example 120/24vac at 5 amps the new one needs to be 5 amps or better. The voltage needs to be the same in this case 24v I chose this as it is a very common size. 

Can you wire them backwards, Yes if wired backwards where the secondary leads are tied to the mains or 120v the out put will be 600v if the insulation holds up (for this example).
Usually step down transformers have a ground wire on the primary side, most of the time they are stamped with the voltage on each side of the case 2 wires of the same color on the 120v side means it doesn’t matter which one connects to the hot and neutral or normally black and white of the 120v supply. 

So make sure your output voltage on the secondary of the transformer is the same as the original and that it is rated for the same load in amps or KW, if the same value is not available go bigger NEVER smaller. And last make sure to hook up the primary side to the supply or mains voltage.