Given a probability space, we say that $(X_t)_{t \geq 0}$ is Markov w.r.t its own filtration $(\mathcal F_t)$ if for all $s<t$,
$$ P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot | X_s)$$
Let $\mathcal G_t = \mathcal F_\infty \setminus \mathcal F_t $. What is the name of the processes that satisfy
$$ P(X_s \in \cdot | \mathcal G_t) = P(X_s \in \cdot | X_t)$$ for $s<t$ ? What is their relationship with Markov processes ? Can you give a non-trivial example of such process ?
If, rather, $\mathcal G_t:=\sigma(X_s: s\ge t)$, then the Markov property as you have stated it is equivalent to the symmetrical relationship $$ P(X_u\in B\mid \mathcal G_t)=P(X_u\in B\mid X_t) $$ for $u<t$. Each is in turn equivalent to the conditional independence of $\mathcal F_t$ and $\mathcal G_t$, given $\sigma(X_t)$, for each $t$.