Across:
1 Responsible for underground movement (SUBWAY) going back (= reversal indicator) on the road (bus = SUB-) to Washington (-WA-) on Fourth of July (fourth of ‘July’= -Y) (SUBWAY),
4 Bull-headed Greek (MINOTAUR) relative (M-A-) admits knocking back (= reversal indicator) most of the remedy (most of ‘tonic’ = toni = -INOT-) from old city (-UR) (MINOTAUR),
9 Is fond of the cast (ANGLER) looking to get to grips with Moby Dick perhaps (ANGLER),
10 Hemmed (STITCHED) in small songbird (-TIT-) caught (-C-) in outbuilding (shed = S-HED) (STITCHED),
12 Tehran bus (Tehran bus = SUNBATHER) broken (= anagram indicator) down on the beach (SUNBATHER),
13 Supporter of the painter (EASEL) in artist’s studio (EASEL),
14 Lack of official sanction (UNACCREDITED) for those not reaching the proper standard (UNACCREDITED),
18 Chew over (PUZZLE) what the chippy has (chippy/carpenter has a JIGSAW tool) for starters? (= position indicator) (JIGSAW PUZZLE) It has a number of pieces and it comes in a box! (JIGSAW PUZZLE),
21 Techie fired for concealing (‘techie fired’ for concealing = CHIEF) big cheese (= CHIEF),
22 Where the police go to catch those looking for a quick getaway (SPEED TRAP) with drugs (SPEED) could be a role (part) reversal? (= reversal indicator) (part = TRAP) (SPEED TRAP),
24 On holiday (OUT-) with potential partner (-DATE-) from department (-D) (OUTDATED) out of Vogue (OUTDATED),
25 10K (-RUN-) in Belgium (B-E-) is one (-I) (BRUNEI) for Malaysian neighbour (BRUNEI),
26 Admit one (admit one = DOMINATE) criminal (= anagram indicator) with influence (DOMINATE),
27 Count (CENSUS) wants to know your religion, for instance (CENSUS).
Down:
1 Doesn’t move (STAYS PUT) remains (STAYS) from place (PUT) (STAYS PUT),
2 Favouritism (B-IAS) over, for example (-EG-), broadcast (-ON-) (BEGONIAS) from the garden centre (BEGONIAS),
3 Earn a (earn a = ARENA) silly (= anagram indicator) reputation for showing off fighting skills and blood sports (ARENA),
5 They get asked lots of questions (INTERVIEWEES) by job-hunters (INTERVIEWEES),
6 Where you’ll find all your options (ON THE MENU) in place of the dinner (ON THE MENU),
7 Taken back (AGHAST) as sandbag has tear within (‘sandbag has tear’ within = AGHAST),
8 It’s difficult to figure out (RIDDLE) what this is? (this clue is a RIDDLE),
11 Police think he did it (CHIEF SUSPECT) for Taoiseach (CHIEF) and that’s not quite right (that’s SUSPECT) (CHIEF SUSPECT),
15 Leader of the clan (CHIEFTAIN) in 21 across (‘chief’ = CHIEF-) ain’t (aint = TAIN) wrong (= anagram indicator) (CHIEFTAIN),
16 In retrospect (= reversal indicator), lawyer (SC = C-S) admits supporter (fan) loses head (f) (‘fan’ without ‘f’ = an = -NA-) joining paramilitaries (IRA = -ARI-) with the last (-Z-) (CZARINAS) of Nicholas and Alexander’s wives (CZARINAS),
17 Loses heart (DESPAIRS) playing (= anagram indicator) dire pass (dire pass = DESPAIRS),
19 Doctor (Dr = -RD), officer (officer commanding = OC = - CO-) and accountant (CA = AC-) flying back (= reversal indicator) (ACCORD) by Concord (concord/harmony = ACCORD),
20 Two men (Vic + Tim = VICTIM) – one of them is the injured party (VICTIM),
23 Some uplifting (= reversal indicator) bootleg ridiculed (some ‘bootleg ridiculed’ = egrid = DIRGE) as a sad song (DIRGE).