Hear no evil, see no evil, speak no evil, clasp hands in front of genitals.
Actually, I’m glad you pointed this out OP.
While everyone is mostly familiar with the three wise monkeys, there’s actually sometimes a fourth one in the mix, Shizaru, which symbolizes ‘do no evil’. This monkey is shown with his arms crossed or with his hands covering his genitals.
So Valve wasn’t just doing that with Soldier to be dirty/silly, they were actually having him represent the lesser known fourth monkey in these monkeys of principle.