Do test centers have clocks?
It will vary. You may want to bring a watch.
i dont think they allow watches.
They don’t allow alarm watches, calculator watches, etc. Good old-fashioned standard watches are allowed (you might need to ask your grandparents what those are. 
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But I agree, while there will generally be a clock, and the proctor will generally announce time remaining, the test taker has the ultimate responsibility for keeping track of time. So bringing a plain watch is a good idea.
DD brought a simple but allowed digital watch that I bought at a dollar store. It couldn’t keep accuracy after a day but still was adequate for the test.
Mine brought a basic digital watch.
My child was told to remove his simple digital watch for the test. He was told that only an analog watch would have been acceptable.
@mairlodi - thanks for the update (mine took SSAT several years ago, so I think we followed guidelines on the website at that time - it was also before internet connected watches were popular).
Not sure about you guys, but when I took my test on the 11th, we had to roll up our sleeves to show we didn’t have watches; no watches were allowed at all! In fact, we weren’t even allowed to use our own pencils, we had to use the ones they provided! Maybe it’s just us…