Standard deviation for a half-normal distribution I have a vector of numbers which is generated from a normal distribution with standard deviation $\sigma$ and $\mu = 0$. When I'm turning these numbers into absolute values, I suppose that the they will follow half-normal distribution with the same standard deviation.
My question is, if I initially start at having a vector of absolute values which generated from half-normal distribution, would it be mathematically sound to add the negative values of these numbers and then find the combined standard deviation between absolute values and negative values to recreate the normal distribution?

If you randomly assign a sign (+ or -) to each member of the sample generated from a half-normal distribution, the result will be indistinguishable from a sample generated from the corresponding unfolded normal distribution.
Your method wouldn't work - for example, if your sample size is 2, it would be very surprising to draw the sample 4, -4 from a normal distribution with mean zero and variance 1, since it would involve two separate 4-sigma events; but it would be much less surprising to generate it using your method from a single sample from the corresponding half-normal distribution, as that would only involve a single 4-sigma event.
