Susu's comment was dead on: the UV editor is the way to address this issue. I'm new to Blender and had never used the UV editor, but read up on it after seeing the comment. For others in the same position, the UV editor allows you to accurately control the mapping of textures to the faces of an object, including mapping different textures to different faces from a single texture map. Here's the approach that worked for me:
- Prepare a single texture file containing all the textures you want to apply (the texture atlas). For example, you could build a cross-shaped graphic with six textures for an unwrapped cube.
- In Blender, press the UV Editing button at the top to split the screen between the 3D Viewport and the UV Editor.
- Put the 3D Viewport in Edit Mode and unwrap all faces of the cube.
- Select each face in turn and use the UV Editor window to select the portion of the atlas that should be used to texture the face, rotating or scaling as required.
This doesn't address the question of why Blender randomly flipped the texture in the first place, but it does make it irrelevant.