go around, rule, administer
斡 centers on the idea of moving around a central point or managing affairs: going around, ruling, or administering. The meanings connect through the image of a hand turning or directing things.
斡 combines 斗 (a measuring dipper or ladle) with a phonetic element that likely contributed the sound. The character is associated with turning or revolving, possibly from the action of a ladle or the idea of managing measures.
The left side looks like a hand holding a tool, and the right side is a measuring dipper 斗. Picture a hand turning the dipper around to stir and manage a mixture, representing going around and administering.
For アツ, imagine the hand turning the dipper so vigorously that it makes a sharp 'ats!' sound, like a quick, decisive action: ats -> アツ.
good offices; kind offices; services; influence; recommendation
people smuggler
job placement; employment assistance
agent's charge
mediation plan; settlement proposal
mediator
mediator; intermediary; agent; broker; middleman
mediated marketing; mediated sales
outplacement