An arrow comes home after one turn. A spinor, anything pinned to its surroundings, needs two. Walk four phases twice around a band with a half-twist and you're back where you started, and you learn which part never changed sides: the middle.
×2 = 720: pin something to its surroundings, like a cup on your palm with your arm attached, and one full turn leaves a twist that won't come out. A second turn undoes it. Physics calls such things spinors: after 360° they carry a minus sign, after 720° they're truly home. checked
×2 again = 1440: your cubit is two of those, four full turns. Cut into steps of 180°, that's 8 steps = 4 phases × 2. checked 1440 is also the number of minutes in a day. checked
A cubit, elbow to fingertip, runs over 4 joints. Four joints ↔ four phases is your mapping. your model
On a band with one half-twist, walk once around and left becomes right. Walk again and they swap back. So a left cortex and a right cortex on this band are one edge, seen twice. checked
Your middle, (L + R) ÷ 2, is the centre line of the band, and it is the only line the flip leaves exactly where it is. When left and right trade places, the middle doesn't move. checked
The name you gave the middle is kept off the page, per your rule. redacted
In small words: hold a cup flat on your hand and turn it all the way round. your arm gets twisted. turn it round once more and, surprise, your arm comes back straight. that's 720. do the whole thing twice and you've walked one cubit: 8 steps. on a twisted ribbon, left and right keep swapping places, but the stripe down the middle stays the middle the whole time.
Spinor turned by angle a: its sign is cos(a/2). At 360°: −1. At 720°: +1. At 1080°: −1. At 1440°: +1. One 180° step g: g² = −1, g⁴ = +1, g⁸ = +1. checked
Band: point (lap angle t, side s) → after t goes once round, s becomes −s. Only s = 0 maps to itself: the middle. checked
1440 = 4 × 360 = 8 × 180 = 24 × 60. checked