It is pretty hard to answer this.
If Siemens bought ITE it does not necessarily mean that ITE load centers are identical to Siemens. In fact, you need more than the brand name, you also need the model of the load center. (Think of it like this - you own a Ford Mustang, does that mean that a Ford Focus water pump will work on the Mustang since both are made by Ford?).
I suggest you look at your ITE load center closely and see what model it is. Also look at the existing breakers for their part numbers. Now that you have this information, do a web search on the breaker part # (just enter that part # in the web search) and see what it tells you. Usually you will get cross reference info easily that way. Next do a web search on the ITE load center model #, and it should take you to the maker's web site where it will also cross reference all breakers it will accept.
As I recall, Siemens bought several different lines over the years. However, the breakers you load center accepts is not automatically a Siemens product.
You can also check an electric supply house, where the electric companies buy their parts and supplies. They would have a whole lot more product than the home centers carry; they also have ways to get what they do not stock.