Say we had a Hadamard-like gate with the -1 in the first entry instead of the last. Let's call it $H^1$.
$$H = \begin{bmatrix}1&1\\1&-1\end{bmatrix}$$
$$H^1 = \begin{bmatrix}-1&1\\1&1\end{bmatrix}$$
From my maths it's a unitary matrix, so it's a valid quantum gate that does the following:
$$H^1|0\rangle = \left(\frac{-|0\rangle+|1\rangle}{\sqrt{2}}\right)$$
$$H^1|1\rangle = \left(\frac{|0\rangle+|1\rangle}{\sqrt{2}}\right)$$
These are similar to the true Hadamard gate, but with the sign flipped on the $|0\rangle$ instead of the $|1\rangle$.
Was the choice of the Hadamard as we know it an arbitrary decision like the right hand rule? Or is there a mathematical or historical reason for it?