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Well, I could have sworn I saw a de blade stropper--a handle like you see for the old Wedge SE blades. Maybe I was wrong?

But I reckon a strop that small, made by Gillette, is almost certainly...

It's probably for stropping DE blades, or something to rival the Valet Autostrop. Gillette and Autostrop were big rivals, and it wouldn't surprise me if Gillette started sniping their products.