Palindromes on the $\tau$-circle: A note for Palindrome Tau Day, 6/28/26

An integer palindrome is a self-reciprocal polynomial evaluated at its base, so its roots are symmetric about the unit circle – where the coordinate is angle, in turns of $\tau$. Read this way, the date $\mathtt{\text{6/28/26}}\to 62826$ secretly contains the primitive cube roots of unity – at angle $\tau \mathrm{/}3$ – along with one further pair of roots on the circle.