We cashed checks for our customer drawn on Bank X. Bank X's customer claims forgery. Bank X paid the checks and wants us to reimburse them. We had no way to verify Bank X's customer's signature. We shouldn't have to pay Bank X, should we? Don't they need to go to our customer for payment?

If Bank X doesn't discover the forged signature within the midnight deadline, they are on the hook.

In some clearinghouses, there are check fraud rules that provide for "adjustments" after the midnight deadline. The clearinghouse rule modifies the UCC in such cases. One excellent example is the check fraud warranty rule at the Central Oklahoma Clearinghouse Association (COCHA), at https://www.cocha.org/Services/CheckFraud.asp

But if you don't belong to a group like COCHA with a similar warranty rule, Bank X (from your example) is "on the hook" if it misses the midnight deadline. It's then a matter between Bank X and its customer (and the forger, if he or she can be found).