Why does pre-increment not work for an overflow check, but post-increment does?

Question

I attempted to write a function that increments an unsigned integer and checks if the result overflowed, by comparing the old value of the integer with the value after incrementing:

unsigned detect_overflow(unsigned x) {
    if (x > ++x) {
        std::cerr << "Overflow occurred: " << x - 1 << " -> " << x << std::endl;
    }
    return x;
}

However, it doesn't work when the comparison in the if is written this way. Calling the function with UINT_MAX leads to undetected overflow.

But if the condition in the if is changed to use post-increment instead:

unsigned detect_overflow(unsigned x) {
    if (x++ > x) {
        std::cerr << "Overflow occurred: " << x - 1 << " -> " << x << std::endl;
    }
    return x;
}

then overflow is detected correctly:

Overflow occurred: 4294967295 -> 0

The same is true for the equivalent version of said function for detecting underflow.

What gives? I understand the difference between the value returned by post and pre-increment operators, but clearly my assumptions about how these values may be used in expressions containing side-effects are incorrect. What is going on here?