Thank you.
More info that I did not have yesterday.
When the valve is open (flow is intended for shower)hot water only flows to the shower. The tub spout gets all the cold.
There is a place on the valve stem that seemed to me to be a place for an o-ring. But there was no residue or any other indication that anthing had ever been there. I asked at the store, and the attendant there said no. I will return and see if I can find an o-ring that will fit.
I looked into the wall with a flashlight, but I am still unsure that I understand how the thing actually functions. In some other post here or at another board, I saw a note (unrelated to my inquiry) that suggested favoring cold over hot would mean that there is a balancing feature, and to look for a screw-like device in the assembly. I looked, but did not see anything like that.