= Hadamard interchange of Y eigenstates
{c}
{title2=$H|\pm i\rangle=e^{\pm i\pi/4}|\mp i\rangle$}
For $|\pm i\rangle=(|0\rangle\pm i|1\rangle)/\sqrt2$, direct <matrix multiplication> by the <Hadamard gate> gives coefficients $(1\pm i,1\mp i)/2$, proving the displayed identities. This exchanges the two <Pauli eigenstates> of the <Pauli Y gate> and agrees with $HYH=-Y$. The phases in the formula are <global phases> for each separate state; the <relative phase> inside each state is essential.
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