One gesture runs through this whole family: turning something over. Flip a thing and it faces the other way, so the noun הֶפֶךְ is the opposite, and the adjective הָפוּךְ is upside-down, reversed, the very word in קָפֶה הָפוּךְ, a "flipped coffee", a latte. Flip a state of affairs and you get a reversal; flip a whole society and you get a revolution, מַהְפֵּכָה. The plain verb even does grammar duty: "to turn into" something. From a coffee order to the overthrow of a regime, it is the same three letters turning over. Every word is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Only the simple pa'al is lexicalized for everyday use, but it is remarkably elastic: "to flip" a pancake and "to turn into" something are the same verb. The reversing meaning is then carried mostly by adjectives (הָפוּךְ, הָפִיךְ) and a rich set of nouns rather than by extra binyanim.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.