Is there a mathematical relationship between a bitpattern and the mirror of that bitpattern?

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For a given bitpattern, is there a mathematical relationship between that pattern and the mirror image of that pattern?

E.g. we start with, say, 0011 1011. This mirrors to 1101 1100.

We can mechanically mirror the pattern easily enough.

But is there in fact a mathematical relationship between the pattern and its mirror?

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Not really an answer, but too long for a comment and should get you started:

Break down the definition of "bit pattern" (really, just the figures that make up a number) to see if there is any relationship.

For a number X representing in base B, its "digits" are the values a_i that make the following equation correct:

X = a_0 + a_1*B + a_2*B^2 + ... a_n*B^n

So for example, in base 10, the number 42 has the following expansion:

42 = 2 + 4*10
a_0 = 2
a_1 = 4

So then let's define the reversal of that number as follows:

X' = a_n + a_(n-1)*B + a_(n-2)*B^2 + ... a_0*B^n
24 = 4 + 2*10 (the reversal of 42)

From this we can easily operate on X and X' to see if there are any interesting relationships. For example,

X+X' = a_n + a_(n-1)*B + a_(n-2)*B^2 + ... a_0*B^n + a_0 + a_1*B + a_2*B^2 + ... a_n*B^n
 = (a_0+a_n) + (a_1+a_(n-1))*B + ... + (a_n+a_0)*B^n

So every "digit" in the sum is equal to the original digit's value in X plus the opposite digit from X. That's kind of obvious when you add something like 42 to 24 to get 66, but less obvious if you were add 67 to 76 to get 143.

I think you'll find that there aren't really many interesting relationships of numbers' reversals.