Consider a Cauchy problem $$\dot x(t)=f(t,x(t)),\quad x(T)=x_T \qquad (*)$$ where $T>0$ and we would like to compute $x(0)$.
I wonder how to construct an ode such that integrating forward the latter would lead to $x(0)$. I don't have precise idea on how to obtain such equivalent ode: I would use time reparameterization, change of coordinates ...
I propose the ode:
$$\dot z(t) = -f(t,z(T-t)),\quad z(0)=x_T, \quad t\in [0,T] $$ then we can recover $x$ from $x(t)=z(T-t)$.
But in the general case (without symmetry on the solutions), this may be difficult to handle numerically, because in numerical implementation, we must provide $g(t,z(t))=-f(t,z(T-t))$ which is not suitable because we can't express (in general) $z(T-t)$ in terms of $z(t)$.
Are there any other odes that could be integrated forward to obtain $x(0)$ ?
Your approach works, because it yields another ODE:
Let $z(t)=x(T-t)$. Then
$$ \frac{\partial }{\partial t}z(t)=\frac{\partial }{\partial t}x(T-t)\\ =-x'(T-t)=-f(T-t,x(T-t))\\ =-f(T-t,z(t)). $$