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Grot is from grotesque and implies dirty or primal, as in grotto, a cave. Sans-serif fonts were called grotesque because they looked primitive. The word grot appears in various 19th century poems as a noun meaning a grotto, where it was supposed to be a lovely place full of fairies. But that meaning wore off and now something grotty is something you'd find in a gutter. The US equivalent is grody.

Tat meanwhile, like gunny, refers to a kind of sackcloth. These words are both from Hindi. The concept is of some cheap filler material. If you want to rush something out, you can wrap it in sackcloth, bag it up in sackcloth, sell it to them in sackcloth, make it quickly out of sackcloth. Or maybe it was left over and has been sitting in a corner for years, at the bottom of a sack. That is "some old tat".