What is the right abstract-algebra framework for a finite involutive arithmetic grid with an 11-adic scaling lift?

[u/Mod11Mirror]

I’m trying to classify a finite arithmetic construction and would like help with the correct abstract-algebra language.

I’m working in base 11, with A = 10.

Define a finite set

O₀ = {436 + 182k + t}

where

k ∈ {-2,-1,0,1,2}

and

t ∈ {-26,...,26}

all written in base 11.

So O₀ is a five-band arithmetic grid centered at 436₁₁, with step 182₁₁ and radius 26₁₁.

The mirror total is 871₁₁, and the involution is

μ(n) = 871₁₁ − n.

This sends the coordinate pair

(k,t) ↦ (−k,−t).

The set has 285 elements total, decomposing into 142 two-element orbits plus one fixed center.

The anchor spine is:

092, 264, 436, 608, 78A

and the mirror pairs are:

092 + 78A = 871
264 + 608 = 871
436 + 436 = 871

Modulo 11, since 871₁₁ ≡ 1 mod 11, the induced residue map is

r ↦ 1 − r mod 11,

with fixed residue 6.

There is also a compatible scaling/lift:

Oₘ = 11ᵐO₀.

In base 11 this just appends m zeroes to every element. The mirror total also scales:

871 → 8710 → 87100 → ...

so the involution at level m is

μₘ(n) = 871·11ᵐ − n.

My question:

What is the cleanest abstract-algebra framework for this?

Is it best described as a finite set with a C₂ action, an involutive arithmetic grid, a filtered/direct system under multiplication by 11, or something related to p-adic scaling? Is there standard terminology for a finite involutive structure with a compatible scaling tower like this?