Thanks for a very entertaining, if somewhat enfuriating, cryptic. I had to "cheat" on some of the clues.
20A20AYoung twins' whirlwinds (8)[--------] Nasty! Seeing Y as the result of "young" I couldn't understand where the initial C came from; of course, I should have remembered the obscure baseball player, that sport being so very popular in the UK!
5A5AiPod engulfs significant other of a pill bug, maybe (6)[------] "Pill bug" is not a common name in the UK.
13A13AReagan depicted a detective (5, 4)[---------] "Reagan" - definition by example?
6D6DThe Reverend's neat ability creates a river (10)[----------] had me mystified! I assume "ability" is SKILL but then I'm lost.
9D9DPair of jingoists leave an Asian capital's core (5)[-----] I assume "Pair of" means the first two letters of "jingoist" but I have never seen this indicator before.
23A23AHot Spanish bed at first tarnished a medal (9)[---------] In the UK, I've never heard of the Caldecott medal! CALDE must be a very obscure way of saying "hot" in Spanish.
24D24DWorkshop or work (5)[-----] Here we go again! US spelling foxes UK solver!
29A29ALeftist leader Michaels flips sign up (6)[------] This one had me tearing my hair out (and there's not much left to tear at)! More US spelling and "Lorne Michaels" - who he?
27D I thought I'd solved the blighter then that US Z appeared! In the UK you are more likely to see S.