Does XOR'ing a random number with its inverse reduce its security if the result is a palindrome? [And four related questions/examples below:]

2) If a random 256-bit number is XOR'ed with its reverse (i.e. the big-endian version XOR'ed with the little-endian version), does this reduce the security properties in terms of bits, if the result is a palindrome, as there are only ${2^{128}}$ such palindromes in the range of  ${2^{256}}$? 

3) Can an adversary efficiently brute-force search just the palindromes in that range? 

Example:  an L-bit random N ${\oplus}$ reverse(N) = palindromic string L-bits long.

Example using a random 256-bit binary string:

N=1000001110111000001110001111100111001000111111111111010110111001011010110110010110011001100010110110101101001000001110111111111111001110101001110001101101110101010101110011000111000010101011001010100000110110111000010101101000100111001010110110011011010110

Reverse(N)=0110101101100110110101001110010001011010100001110110110000010101001101010100001110001100111010101010111011011000111001010111001111111111110111000001001011010110110100011001100110100110110101101001110110101111111111110001001110011111000111000001110111000001

N ${\oplus}$ Reverse(N) = 1110100011011110111011000001110110010010011110001001100110101100010111100010011000010101011000011100010110010000110111101000110000110001011110110000100110100011100001101010100001100100011110100011010110011001000111100100100110111000001101110111101100010111

4) Are there any advantages in using a longer key with reduced security than a smaller key with the same security (i.e. could there be an advantage having a 256-bit key with 128-bits of security, compared to having a 128-bit key with a 128-bits of security?)

5) In addition, could there be benefits such as only having to retain half of the palindrome, in order to reconstruct the other half, in terms of compression/notation while avoiding information loss despite any reduced security?

Example using above palindrome where only the leading 128-bits are retained:
 11101000110111101110110000011101100100100111100010011001101011000101111000100110000101010110000111000101100100001101111010001100 and where reconstruction of the original 256-bit palindrome is achieved by concatenating the reverse of the 128bits to the right-end of the retained 128 bits as follows: 11101000110111101110110000011101100100100111100010011001101011000101111000100110000101010110000111000101100100001101111010001100||00110001011110110000100110100011100001101010100001100100011110100011010110011001000111100100100110111000001101110111101100010111
The xor of two random strings is a random string, so you're basically generating a 128-bit random string from a 256-bit random string.


Yes, it reduces security compared to a pure 256-bit random string.
Yes. As you pointed out, there are only $2^{128}$ possible strings.
Yes. The attacker can generate all $2^{128}$ strings and trivially concatenate them with their reflection in each step.
No, both have 128-bit security against brute force attacks.
No, as mentioned there is no security gain in using palindromes, which implies that there is no advantage at all in use palindromes. For example, a 256-bit palindrome can be stored "compressed" in 128 bits, but it still provides only 128 bits of security. You could simply use a normal 128-bit key instead, which fits in the same size, has the same security, and is a lot simpler.