Very nice! The categories are
ACHIEVE AUDIOTAPE APHID VEGETABLE all contain a Greek letter (CHI, IOTA, PHI, ETA)
AHA EON LADA UPON all are Greek letters missing a middle ALPHA EPSILON LAMBDA UPSILON)
AQUATIC MISPRINT SIXTH SKUNK all contain the reverse of a Greek letter (TAU PSI XI NU)
AZTEC HOMAGE IMAGES OMNICORP are all anagrams of Greek letters with an extra letter (ZETA OMEGA SIGMA OMICRON)
HAPPY JAMMY MELTS WHETS are all Greek letters with their first and last letters changed (KAPPA GAMMA DELTA THETA)
MIAOW TART TIER WHISK are all synonyms of homophones of Greek letters (MU/MEW PI/PIE RHO/ROW BETA/BEATER)
The hidden theme is
... clear, but worth noting the title. In series one (and two and three) of "Only Connect" the question choices were Greek letters before switching to hieroglyphics in later series.
with an extra tip of the hat
for using every letter exactly once.