How to properly time-shift a function in relation to another function
See that if you are going to go from $y(t)$ to $y(t-\tau)$ then you should replace every $t$ in the first one with $t-\tau$. This means that, in your case,

$$y(t-\tau) = x(t-\tau-2) + x(2-(t+\tau)) = x(t-\tau-2) + x(2-t+\tau)$$

In other words,

$$y(t) = x(-t)\implies y(t - \tau) = x(-(t - \tau))= x(-t + \tau)$$