Reversibility of an autonomous differential equation ($F(Rx)=-DR(x)F(x)$)

A reversing involution $R$ maps a trajectory $x(t)$ to another trajectory $Rx(-t)$. The chain rule gives the displayed condition. In contrast, an ordinary equivariance symmetry satisfies $F(Sx)=DS(x)F(x)$ and maps trajectories with time unchanged. Oddness $F(-x)=-F(x)$ expresses half-turn equivariance, not reversal by the half-turn. Damping usually breaks the mechanical candidate $R(x,y)=(x,-y)$.