I can confirm that such incidence happens with Jazz Jackrabbit 2 1.23, JJ2+ 5.6 via Lutris (Wine-Staging 3.10 i386) on Linux Mint 19 (x64) Mate with Logitech F310 Xinput/DirectInput. As the game is started with Jazz2.exe -menu -windowed -nolog -nospy executable and arguments, mouse cursor immediately moves left even in OS level. When a map is started, the character looks up as it is shown in previous posts. I found related lines in the readme of the latest release of JJ2+:

"Improved support for DirectInput/XInput (read: XBox-style) controllers:

JJ2+ should now ignore corrupted controller data instead of seeming to constantly press the left and up keys."

https://docs.jj2.plus/

It looks like some Windows and WINE setups fail to "gnore corrupted controller data." When -nojoy argument is added, problem disappears but as it disables controller support, it's a bit pointless. Alternatively, Linux users can emulate keyboard buttons on controller with AntiMicro but that would be a dirty solution to be honest.

Here are the Lutris page and code of the installer script:

https://lutris.net/games/jazz-jackrabbit-2/
https://lutris.net/games/install/2782/view