Binary Palindrome
A different mirror, a different kind of symmetry.
A number whose binary representation reads the same forwards and backwards, like 5 = 101₂ or 27 = 11011₂. The decimal appearance is irrelevant — 27 looks ordinary, but in binary it's perfectly symmetric.
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Binary and decimal palindromes are two nearly independent lotteries: from 0 to 1,000,000 there are exactly 2,000 binary palindromes versus 1,989 decimal ones — almost a tie in count, yet their overlap is tiny. An ugly duckling in decimal may be a swan in binary.
Why exactly 2^⌊(k−1)/2⌋ binary palindromes of k bits? The leading bit must be 1, and only the first half of the bits are free — the rest are locked by mirroring. Same "half the degrees of freedom" principle as decimal palindromes, just in base 2.
Trivia: every binary palindrome above 0 is odd (the top bit is 1, and mirroring forces the bottom bit to 1), and every even-length binary palindrome is divisible by 3. Powers of 2 never appear on this list.