On single ended inputs the signal goes both positive and negative with respect to the chassis. The preamp does not "generate" a missing negative half. There is nothing missing that needs to be generated and nothing gets split.
The job of the preamp is to amplify the input voltage and to have a low enough output impedance to drive the cables and the input to the amp. There are three ways to configure a tube. Common plate is commonly callled a cathode follower
Z means impedance.
common grid = input on cathode and output on plate
common cathode = input on grid and output on plate
cathode follower = input on grid and output on plate
======================= voltage =============== inverts
============= input Z ==== gain ====== output Z === polarity
common grid======= low ===== high ====== high ======= yes
common cathode === high ====== one ====== low ======== no
cathode follower ==== high ===== high ====== high ====== yes
ideal preamp ======= high ===== high ====== low ========= no
Overall the preamp must have a high input Z, high gain, and low output Z. As you can see none of the three configurations meets this requirement. To get around this problem, in a typical minimalist preamp you have two stages. The first is a common cathode which gives you voltage gain but inverts polarity. It also has too high an output Z. So this is coupled to a cathode follower that gives no gain but lowers the output impedance. The signal reamins inverted. To get it back to the original polarity would require another inversion. A lot of designers don't want to add another stage so they leave it inverted.
There are other design schemes but this is probably the most common used. Sorry for all the ======== but if you use more than one space in a row it gets reduce to one space.
Note to Audiogon: You have one of the most frustrating word processors I have ever used for posting ads and chat. Why can't you just leave the puncuation and capitalization the way we enter it???