Yester-day and the day before, I listened to the two discs of an anthology of the Tremeloes titled Silence Is Golden: Anthology. I noticed that the riff in "Shake Hands (and Come out Crying)" is really similar to that in the Byrds' "I'll Feel a Whole Lot Better." And then I remembered that the riff in that Byrds' song came out of the Searchers' "Needles and Pins." Between the three, I think there's a difference of only one or two notes. They're all in the same key too - A major.
Not much is said about "Shake Hands (and Come out Crying)" in the liner notes to this anthology. Really, the only relevant detail I can find is that it's from 1967 - a few years after "Needles and Pins" and "I'll Feel a Whole Lot Better." So while it's possible that the riff in "Shake Hands (and Come out Crying)" was taken from either "Needles and Pins" or "I'll Feel a Whole Lot Better," I'm not sure if it was. It's also possible that it's just a coincidence, but it's too similar for me to think that it's likely.
If it was taken from "I'll Feel a Whole Lot Better," it'd be an interesting instance of inspiration hopping across the Atlantic - first the Byrds' taking that riff from the Searchers' "Needles and Pins" (an American band taking inspiration from a British band), and then the Tremeloes taking it from the Byrds (a British band re-claiming that inspiration from an American band). But - again - I don't know where the Tremeloes got the riff.