Three consonants hold one social event: joining two families through marriage. The same letters give you the groom, the wedding, the act of marrying someone off, and the act of getting married yourself. Notice the side meaning hiding in plain sight: חָתָן is both the bridegroom and the son-in-law, because to marry into a family is to enter it. Every word here 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.
The contrast here is the heart of the root. Pi'el לְחַתֵּן is what a parent or matchmaker does to others (marry them off); hitpa'el לְהִתְחַתֵּן is reflexive, what you do to yourself (get married). No pa'al verb is lexicalized for this root; the family lives almost entirely in pi'el, hitpa'el, and the nouns.
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.