The texture you're talking about comes from the screen on which the wet paper slurry is poured, allowing the majority of the water to drain away. It comes in two basic flavours: laid (most of the wires in the screen go only one direction, usually horizontally across the page, with only a few wires going the other direction) or wove (more like window screening). The rest of the process involves squashing the formed sheet of paper flat, removing that screen texture to one degree or another. It's not something that's impressed in the paper after it's a sheet.
The only standard papers that have a definite texture and more-or-less the same texture on both sides are rough and cold pressed heavy papers, such as you'd use for watercolours. That texture comes from felt blankets used in the wet squeezing process, and is quite irregular and lumpy rather than the regular texture of a laid or wove screen.
You can get decorative craft and cover papers that have textures on both sides, but they tend not to be white-ish and not terribly printer-friendly.