I have a new switch that requires a neutral. There is a neutral wire in the box, in the back, but it's continuous -- comes in one side and goes out the other. Is the proper procedure to cut it, then bundle the two ends along with the neutral going to the switch? And then cap the new bundle of course.
You are in conduit, which changes everything.
Nope, you can't cut it. You don't have enough length remaining. Both stubs will be illegally short and you would then have to replace both of them.
However, what you can do is replace just one half of the run, left or right, whichever one is easier to fish/pull. Continue using the original wire on the other half of the run.
Identify which one is easier to replace by opening up both connecting boxes, and push and pull on the white wire from here. You will get tactile feeling on which one moves easier. While you're there, see if there is an abundance of slack in those boxes and you can just move wire around. The legal absolute bare minimum is 3" of insulated wire sticking out beyond the surface of the wall at each box.
Then on the "easy end" pull the slack out of the old wire, and lash it to the new wire with electrical tape at least 6-8", as long as you can. Streamline the "nose" so it doesn't hang up on things. Then, from this box, pull the old wire out, dragging the new wire in.
Shove the old wire down the other pipe back to center, because you probably pulled all the slack out of the other box. That box needs that slack. Then unsplice old and new wire, position everything and cut all the wires so they are 8" beyond the surface of the wall.
You could also leave the old wire and double back from the next or previous box on that conduit. I do the "pass thru and double back" trick all the time to reduce practical box crowding, but that won't help you here.
Since you will be making all your joins at a wire nut, you might as well use stranded THHN wire which is more flexible and easier to handle, there's also no stiffness penalty for #12 wire. Must be white or gray.