The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
VENUS, NEBULA, METEOR, and ORION all fit the category "Night sky". Look up after sunset. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Card suits and kin
CLUB / DIAMOND / SPADE / HEART
CLUB, DIAMOND, SPADE, and HEART all fit the category "Card suits and kin". A deck has all four. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things with shells
EGG / ACORN / TURTLE / CLAM
EGG, ACORN, TURTLE, and CLAM all fit the category "Things with shells". Each can have a protective outer layer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things you fold
LAUNDRY / MAP / LETTER / ORIGAMI
LAUNDRY, MAP, LETTER, and ORIGAMI all fit the category "Things you fold". A crease is part of the process. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.