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Some folks have had good luck with seat belt material for the fabric component. If you have a junk yard nearby you could try a length of that stuff.

I've read of people attaching their strops to a towel rack but it seems to me that would be an awkward angle to strop a razor. OTOH, some do it and get along fine. I like a door knob because it is...

If you were able to come by the traditional hardware (clamp & swivel) that strops normally have that might alleviate the twisting issue that you have. JimR in Japan sells strops and might be able to...