I thought the same thing as well. It's hard to believe the similarity between rotate and tate is just a coincidence.
What a shock?
I'm not too much man to admit I'm wrong:
縦
1. 〈長さ〉 (a) length; 〈高さ〉 (a) height
縦30センチ横20センチ高さ10センチのブロック a block 30 by 20 by 10 centimeters
縦の lengthwise; 《フォーマル》 longitudinal; 〈垂直の〉 vertical; perpendicular
(クロスワードの)縦の1番 number one down
縦に lengthwise; lengthways; 《フォーマル》 longitudinally; 〈垂直に〉 perpendicularly; vertically
縦に並べる arrange 《the blocks》 lengthways [lengthwise]; place 《the desks》 one behind another [the other]
縦に線を引く draw a perpendicular [vertical] line
首を縦に振る agree; nod one's agreement; 《フォーマル》 nod (in) assent; 《フォーマル》 give one's consent 《to》
横のものを縦にもしない ⇒
横
1. 〈側面〉 the side; the flank; 〈幅〉 the width
縦20センチ横15センチの箱 a box 20 by 15 centimeters
横の 〈縦に対して〉 《フォーマル》 lateral; horizontal; 〈かたわらの〉 beside; by the side 《of》; by
Cognates FTW.