Patterns in the digits of triangular numbers

Here's another way of saying what I said in my post on congruence patterns in the series of triangular numbers.

List the triangular numbers, starting with 0. The final digits of the numbers in the series will exhibit a repeating pattern with a period of 20, and the pattern will be palindromic. The penultimate (tens-place) digits will also exhibit a repeating palindromic pattern, with a period of 200. The repeating palindrome for the hundreds-place digits will have a period of 2000, and so on for any "place" you care to choose.

This is assuming you write the numbers in the familiar decimal system -- but any number base will yield corresponding results. If the triangular numbers are written in base b, the pth-to-last digit will exhibit a repeating palindromic pattern with a period of b^p (if b is odd) or 2b^p (if b is even).

(Oswald Wirth, whose writings first alerted me to the existence of such patterns, was doing what he called "Theosophic reduction" -- i.e., adding up the digits of a number until a single digit is arrived at. "Theosophically reducing" a number in base b corresponds to taking its final digit in base b - 1.)

The patterns are easiest to see in very small bases such as binary and ternary, shown below with color-coding to highlight the repeating palindromes.

I have still not figured out why these patterns exist. Even stating the pattern algebraically is proving somewhat difficult for someone whose mathematical training is as limited as mine.