If it’s a texture you want available every time you start ZBrush, put it in the ZBrush 4\ZStartup\Textures folder. Up to 25 such items can go there.
Alternatively you can put it in the ZBrush 4\ZTextures folder. Then you can use the Textures tab in LightBox to view the thumbnails of that folder’s contents and load the one you want.
Also, if the texture is applied in the Tool>Texture Map menu, it will be saved as part of the Tool. (Tool>Save As for the ZTL format or File>Save for the ZProject (ZPR) format.) Then it will automatically load and be applied when you load your tool or project. It won’t clutter the Texture palette either (unless you then use the Tool>Texture Map>Clone button to copy it into the Texture palette).
Regarding why flipping the texture doesn’t affect the Tool>Texture Map menu, it’s because that menu contains an instance of the texture rather than the original. Changes in the Texture palette affect the original, but won’t affect any instances of it that are actually applied to the tool. This allows you to try out various effects without committing to anything until you’re ready.