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I think I get the connection between the gimmick and the theme, but I could be overreading it... on a couple of levels.
Found the clues challenging (but in a good way) and the endgame quick.

Cold solved a few but no idea how to decide which answer goes in which half. And no idea how to start gridding with any certainty. Is there a tactic Drxx or do I just go ahead and start guess-plotting some of the down answers and hope things become clear…?

I haven’t completed this yet, but a wild stab led me to insert the answer to first half of across clue 1 and has enabled the left side to more or less fill itself! So have a go H1982. It was because I could see a shorter word…..But Drxx is your man. I’m such a novice really.

Yes, pretty much trial and error for me too - and a lucky first guess (also with a shortened word in mind).
There are some intriguing alternatives for both longer and shorter entries and I had three corrections to make before the two strands of letters made complete sense.

Having got started with 1a on the left, things fell into place and the first three or four removed letters leads pretty smartly to knowing what the added letters are I thought.

However, I can't satisfactorily parse either part of 20d. (It wouldn't be normal if ever finish a crossword without at least 1 ? 😄

I struggled with the 2nd part of 20d, granama1, the best I can do is -

A homophone of 'base', with the first letter [the 'upper crust'?] leaving to give a cruder slang term for '(the) behind'.
Pretty sure the first part is a synonym for '(secret) agents' - but only the insides/'stuffing'.

Your tip regarding what the letters spell out is helpful - I realised I could have saved myself some time if I hadn't left it until I'd completed the grid before working it out.

The upper NE corner proved to be my LHS 21a (before mutilation) for a very long time, but all is now complete. Back to grid staring at the listener, if I can be bothered.

drxx, wrt the second part of 20d, if "caught" was "heard" in the clue, if confirms your parsing of the answer.

Yes, I'm OK with 'caught', gitto - it's 'upper crust' for the first letter that I find a bit of a stretch (but I'm warming to it}.