-times, round, game, revolve, counter for occurrences
回 centers on circular motion and repetition: something that revolves, goes around, or occurs in rounds. The counter sense is a natural extension of counting occurrences as cycles or turns.
counter for occurrences
this time; now
to turn; to rotate; to revolve; to spin
restoration; recovery; rehabilitation; return; improvement
previous time; last time; previous installment; previous instalment; previous session
rotation; revolution; turn; spin
every time; each round
collection; recovery; withdrawal; retrieval
reply; answer
first time; first innings; initial attempt
mawashi; belt; loincloth
number of times; frequency; count
evasion; avoidance
next time (occasion)
to turn; to rotate; to spin; to twist; to gyrate
circuit; line
to exceed (esp. figures: profits, unemployment rate, etc.); to surpass; to be more than; to be better than
circuit
event with ... rounds, innings, legs, etc.
going around; patrol; round; tour
withdrawal; retraction; revocation; repeal
return (to); revolution; recurrence
revolution; rotation; turning; circling; swiveling
putting off; postponing
回 is a pictograph of a spiral or revolving motion, with the modern form showing a smaller inner 口 inside a larger outer enclosure 囗, suggesting something turning within a boundary.
Picture a spinning top inside a box: the outer 囗 is the box, and the inner 口 is the top whirling around. Each spin is one round, one time, one occurrence.
For カイ, imagine a kite flying in circles: kite -> カイ, and each loop is one round.