Is Kalman smoother symmetric in time?
Running Kalman smoother is not same thing with running Kalman filter backwards in time. They are different. In particular, RTS smoother does not take care of observations, it uses filtered estimates. Kalman filter is an online estimation algorithm and estimates the filtering distribution $p(x_t | y_{1:t})$, i.e., the posterior distribution over the hidden states given the observations up to time $t$. However, smoothing recursions give the posterior of the states given the whole observations: $p(x_t | y_{1:T})$.

Even if your observations are reversible in time (if that is what you imply by saying 'symmetric'), in that case whenever you process the filtering algorithm, the smoother will work backwards and since filtered estimates are incomplete and does not know future observations, they will typically be different.