A series in powers of $(a-z)$ instead of $(z-a)$

Sometimes it is more convenient to find a series expansion (e.g., Taylor, Laurent, Puiseux, ...) in powers of $(a-z)$ than in powers of $(z-a)$. For instance, the command

Series[(1 + Sqrt[1 - 4*z^2])/(4*z) - z, {z, 1/2, 3}]

gives $$i\sqrt{z - \frac{1}{2}} - 2\left(z - \frac{1}{2}\right) - \frac{3}{2}i\left(z - \frac{1}{2}\right)^{3/2} + \frac{23}{8}\left(z - \frac{1}{2}\right)^{5/2} - 4\left(z - \frac{1}{2}\right)^3 + O\left(z - \frac{1}{2}\right)^{7/2}.$$

The series expansion in powers of $\left(\frac{1}{2} - z\right)$ is nicer in this case, as the coefficient at the leading term becomes real. So I would like to ask what is the preferable way to obtain an expansion in terms of $\left(a - z\right)$ instead of $\left(z - a\right)$. Thank you in advance.