FINISHED GRID

The Secret Sauce
Even though the clues are rotated, the letters are NOT! (Super cool, BTW...amazing this works.) So an E can be an M or a W from another direction, O and X are the same either way, H/I and N/Z pair upon rotation by 90 degrees, and C can rotate to U, and vice versa, with the appropriate rotation. So the grid above has hand-drawn characters at intersections to try to capture this.
ACROSS
1. MAW (Where one eats) = M_ A_ W_
2. ELI (Bulldog, as in member of Yale community) = (-m)ILE(-s)<
3. ENZYMIC (perhaps causing a breakdown) = ENZ(-o) + Y (young) + M_ I_ C_
4. WAC (old soldier) = "WHACK"
5. NEIGH (Like Sir Winston call) = _N EIGH_
6. MBZ (head of 19-across) = MetaBoliZes
7. HEX SIGN (barnstar) = tH_ bE_ eX_ ...
8. HRU (welfare check, [how are you]) = HR (hiring department) + U (ripped up)
9. HBO (&lit) = HoBsOn (choice network from "Hobson"), and network where Hobson's Choice could be shown
10. OBE (&lit) = _O BE_
11. CREDO (belief) = C (confront initial) + ODER ("runner" [river] in Police, Poland)
12. URN (place for an old pet) = RUN*
13. MOXIE (some go) = XI + MOE
14. OVIFORM (like a human head) = _OVI FOR M_
15. MAINE (&lit) = MINE + _A
16. HINGE (&lit) = _HING E_
17. HEINZ (mustard manufacturer) = HZ (cycle, abbr for Hertz) + EIN
18. CYMRU (country) = CRUM(-m)Y*
19. UAE (Gulf destination) = sUbArEa
20. COZ (bro) = C (around, circa) + OZ (Animal performer)
21. UFO (something flying through the air) = beaUFO_ (4 and on...clever!)
22. CHORIZO (some sausage) = stomaCH_ + malodOR_ + salamI + protoZO_ (sixth and seventh letters of remaining words...clever!)
23. OLM (animal from the former Yugoslavia) = (-d)OLM(-a)
24. CHE (ddef. Italian what, he moved...)
25. UGH (gross) = aUGHt
26. EVOKE (produce) = OK (sound, as in firm) + EVE
27. WOW (ddef. Amazing and World of Warcraft)
28. ELU (&lit, abbr for English language and usage) = _E + _LU_
Hat Tips
To Gareth McCaughan for tips to get full understanding of 11 and 28
To msh210 for fixing up #20