Yes, the many-worlds interpretation is supposed to be time symmetric.
Consider this toy example with a particle than can be in either of the states $\left|1\right\rangle$ or $\left|2\right\rangle$. Additionally, we denote the no particle state as $\left|0\right\rangle$.
The particle gets emitted by our source $S$ that can be in its ground state $\left|S_0\right\rangle$, in a state $\left|S_1\right\rangle$ ready to emit a particle in state $\left|1\right\rangle$ or in a state $\left|S_2\right\rangle$ ready to emit a particle in state $\left|2\right\rangle$.
Eventually, the particle gets absorbed by a detector $D$ that can be in its ground state $\left|D_0\right\rangle$, in a state $\left|D_1\right\rangle$ after absorption of a particle in state $\left|1\right\rangle$ or in a state $\left|D_2\right\rangle$ after absorption of a particle in state $\left|2\right\rangle$.
Let the time evolution of the states of interest be given by the discrete steps
Note that this is not how unitary time evolution works in quantum mechanics, and I'll get back to that at the end.
So, what happens if we start out in a state $\left|S_1\right\rangle + \left|S_2\right\rangle$?1
This is symmetric in source and detector and time symmetry is manifest. However, in our subjective experience, we only ever see detectors in a state $\left|D_1\right\rangle$ or $\left|D_2\right\rangle$ - we have no concept of a detector in a superposition of states $\left|D_1\right\rangle+\left|D_2\right\rangle$, its pointer indicating two values simultaneously.
Some people argue (or at the very least, have done so historically) that the state must have collapsed through one of the (non-unitary) physical processes
\left(\left|1\right\rangle+\left|2\right\rangle\right)\otimes\left|D_0\right\rangle \longrightarrow \left|0\right\rangle\otimes\left|D_1\right\rangle
\left(\left|1\right\rangle+\left|2\right\rangle\right)\otimes\left|D_0\right\rangle \longrightarrow \left|0\right\rangle\otimes\left|D_2\right\rangle
Others claim that the 'collapse' is merely apparent, a Bayesian update of our information about the world, and the quest for ways to achieve objective collapse is indicative of a fundamental misunderstanding of the nature of reality as revealed by modern physics.
Proponents of the MWI argue that the most sensible resolution is to consider the observer (including their subjective state of mind) as intrinsically liked to $D$, ie we can consider the detector with its macroscopically distinct pointer states as a proxy for the relevant parts of the environment. The asymmetry arises because the observer was blind to the microscopic state of the source, but sensitive to the macroscopic state of the detector, and a time reversal would include a process of memory erasure.
It should be noted that the toy model I presented is cheating, which is why I had deleted my answer:
Emission and absorption are not just given by unitary time evolution like
U(t_i, t_f) \left( \left|S_1\right\rangle\otimes\left|0\right\rangle \right) = \left|S_0\right\rangle\otimes\left|1\right\rangle
Instead, we're dealing with stochastic processes (the source could in principle stay in its excited state indefinitely), and there's already a 'collapse' implied by the discrete time evolution of the toy model that was glossed over.
However, I suspect the line of reasoning I presented here would make sense in a consistent histories approach to the many-worlds interpretation.
For now, I'll leave the answer as-is - I'm not sure I can come up with a better one without a few sessions of late-night (or beer-fueled) philosophizing...
1 we ignore issues of phase and normalization