I believe if you look in the panels around the mobius double reacharound establishing there's something that looks like the cardinal movement in the BG of aradia's land

That’s the issue though-

There’s not one, there’s more than one. Music Boxes and Quartz Crystals are staples of Aradia’s Land, the Cardinal Movement is just a significantly bigger Music Box, and there’s not much allusion to it. In fact, ‘Cardinal Movement’ is mentioned exactly one time in the comic, and it’s by Aranea. Which is what makes me think Aradia’s Land didn’t exactly have a Scratch Thingy.