My understanding is that at the level of quantum mechanics, almost all operations are reversible in time. Most gates in a quantum circuit clearly obey this rule; they can be reversed by applying some other gate. But a measurement gate seems to be one-directional. Is there some physical process that performs the opposite of what a measurement gate does?
A measurement is basically a CNOT between the quantum computer and the external environment. The important distinction between this CNOT and the CNOTs entirely within the quantum computer is that the target qubit is not protected. The environment is going to spread and mix the target qubit's value all over the place.
To perform the inverse of a measurement would mean you need to undo all the mixing and spreading that the environment does, so you can undo the CNOT at the heart of the measurement. Unfortunately this is not plausible, because you don't have sufficient control over the environment.
The mixing is so hard to undo that some interpretations of quantum mechanics say it triggers a literally irreversible process (collapse). But even in interpretations without collapse, like many worlds, it's not practical to undo the mixing.
A measurement leads to collapse of a wave function describing qubits, they get to one particular state and remain in that state. To do reverse operation you have to repeat measurement many times. Based on resulting probability distribution you can reconstruct the state of qubits and prepare it again.
The reconstruction of probability distribution is done with quantum tomography.
Arbitrary state can be prepared with method in this paper: Transformation of quantum states using uniformly controlled rotations
If you look at quantum circuit diagrams, you'll usually see quantum rails (single-lines) and classical rails (double-lines). These rails represent quantum bits or classical bits. The measure operator is performed on a quantum bit, but outputs to classical space. So it's NOT a quantum operator. It does not need to be reversible.