By using the CLI on Linux, can I make a running blender instance save its current scene to file? I have a desktop machine that has been baking a physics (rigid body) simulation. I am now from another computer, and I have ssh access to the remote one. In the remote /tmp folder the last autosave is from before "bake" was invoked. 

Is there a way to send a signal in order to make the existing blender instance save its current scene to file, or to trigger its autosave?
No. Blender accepts cli options when starting that can include a python script to automate tasks but once started it no longer accepts commands from the cli. 

The autosave, if is hasn't been disabled, should save a temporary copy, even the unmodified startup file will regularly autosave a temporary copy. If your autosave file is old then it may indicate that the copy of blender is no longer running.

If you can't get back to the machine (or get someone else) to close it properly then the only option I can think of trying is x11vnc that can connect to an existing xorg desktop. See if you can download a copy to your home folder and run it without admin permissions.