How do I stop thin steel tube from twisting while welding a tall bookcase frame?

The shift you're seeing is heat distortion, and on thin square tube it's almost unavoidable if you weld one side fully before moving on. The weld shrinks as it cools, and it pulls the joint toward wherever the bead is. That's the actual mechanism behind everythign people tell you about stitch welds and walking around the piece.

What worked for me on a long frame: tacking does most of the damage, so treat tacks like mini welds. Put one tack, then go to the opposite corner or opposite face and put one there. Alternate every time. A tack pulls maybe half a degree per joint, but a dozen on one side adds up fast. On welding order, the big thing is balancing each joint. Tack everything fully assembled before you run any real beads, becasue the tacked frame is what locks your geometry in. Then when you weld, do one short stitch, move to the opposite side of the same joint, do the same length stitch. Skip around the whole tower between passes. Never finish a joint in one go. Also, don't trust the flat table to keep you square. Clamps hold things in place while the metal is hot, but the moment you release a hot joint it moves anyway. Check square with a diagonal measurement across the whole face, corner to corner, not just with a square at each corner. If the diagonals match, the face is square, and you'll catch distortion way earlier. As for dead straight: no. Not on a 2.5 metre thin tube frame, not without fixtures. I had a buddy in a proper shop build something similar and even he had to shim and adjust at the feet. Build some slotted holes or adjustable feet into your base plate design and let the floor take care of the last few millimetres. Honestly that's what commercial fabricators do too, so nobody will think less of the piece. Last thing, if it does pull out of square despite all this, you can sometimes bend it back cold. Thin tube moves more than you'd think with a bar clamp and a straight edge across the face. Do it before final paint and you'll never see it.

One trick that saved me on a gate frame last summer: bolt a length of straight box section right across the face as a brace while you weld. The frame physically can't pull out of square with a rigid member fighting it. I used some offcut of 50mm tube and two bolts, took ten minutes to set up. Pull it off after everything's cool and the geometry is locked in. Also cut your weld size down, a fat fillet on thin tube just dumps more heat into the joint for no strength gain, so a smaller leg length does the same job with way less pull.

I'd push back on the rigid brace across the face idea. A brace locks whatever face you bolt it to, sure, but a tower like this racks in the other plane too, and the unbraced faces can still pull while you're welding. Worse, if you clamp a frame that's slightly out against something dead straight and weld it that way, it springs the moment the brace comes off. I've seen it happen on a table base I built. Better to tack it true on its own, check the diagonals, and use the brace as a second oppinion, not the thing holding your geometry.

One thing that helped me more than anything on a similar tall frame was just letting it cool completely before I touched it. I kept flipping and repositioning mine while joints were still warm and every move added a new twist. Now I walk away for a coffee between steps and it's made a real difference. Also, what process are you using? If you're on MIG you can usually drop the wire speed a touch and it'll pump less heat into thin tube per pass, which makes the pulling way more forgiving.

Update: I really focused on the advice to let the frame cool entirely before making the next movement. I kept myself from flipping the section or touching it for a good twenty minutes after each weld. It sounded like overkill, but once I did it, the frame stayed much tighter when I started the next row of joints. The heat distortion was still there, but it wasn't compounding the issue anymore.