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.
SOAP, SIGNAL, MUSIC, and CHOCOLATE all fit the category "Things with bars". Bars can describe each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Musical directions
PIANO / ADAGIO / ALLEGRO / FORTE
PIANO, ADAGIO, ALLEGRO, and FORTE all fit the category "Musical directions". A score might tell a player these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Baking staples
FLOUR / YEAST / BUTTER / SUGAR
FLOUR, YEAST, BUTTER, and SUGAR all fit the category "Baking staples". A bakery keeps these close. 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
TURTLE / CLAM / EGG / ACORN
TURTLE, CLAM, EGG, and ACORN 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.