Whitey spent the whole day at the vet yesterday, waiting for his kidneys to produce enough urine for a useful sample. Once they finally did, it turned out that he has no active infection and no crystals in his urine. Yay!
Thus, his on-again off-again litterbox troubles are attributed to interstitial cystitis, a recurring bladder inflammation. He's been given anti-anxiety medicine, and we've been given instructions to call the vet in a month and keep a log of how he feels.