Aries, Cancer, Libra and Capricorn are the four signs the Sun enters at an equinox or a solstice. Aries and Libra begin at the equinoxes, Cancer and Capricorn at the solstices, and between them they mark the four moments at which the year changes direction.
Ptolemy calls them the tropical signs, from tropikos, a turning point, and the name is astronomical before it is symbolic. It records what happens: the Sun's declination reverses at the solstices and it crosses the celestial equator at the equinoxes, so a season begins in each of these four signs and in no others. The initiating character the tradition assigns them follows from that fact rather than from a separate doctrine laid on top of it.
The four are distributed one to each element and one to each polarity in strict alternation. Aries is fire, Cancer water, Libra air and Capricorn earth, and they alternate active, receptive, active, receptive round the circle. That is a general property of the zodiac's construction rather than a special feature of this group, but it does mean that emphasising cardinality in a reading is not thereby emphasising any one element.
The modern names for the mode come later than the Greek one and carry a slightly different sense. Tropical says the season turns here. Cardinal says the sign leads.