A gate that returns the original wavefunction. $\hat{I}\ket{0} = \ket{0}$ and $\hat{I}\ket{1} = \ket{1}$

$$\begin{align} \hat{I}\ket{\psi} &= \hat{I}(\alpha \ket{0} + \beta \ket{1} ) \\ &= \alpha\hat{I} \ket{0} + \beta \hat{I} \ket{1} \\ &= \alpha \ket{0} + \beta \ket{1} \\ &= \ket{\psi} \end{align}$$

A gate that rotates 180 degrees about the +x-axis on the Bloch Sphere. $\hat{X}\ket{0} = \ket{1}$ and $\hat{X} \ket{1} = \ket{0}$

$$\begin{align} \hat{X} \ket{\psi} &= \hat{X} (\alpha \ket{0} + \beta \ket{1} ) \\ &= \alpha \ket{1} + \beta \ket{0} \end{align}$$

The same effect (switching) occurs in the $i$ basis, since the reflection also affects the $y$-axis.

Rotates 180 degrees about the +y-axis. $\hat{Y}\ket{0} = i\ket{1}$ and $\hat{Y}\ket{1} = -i\ket{0}$

Rotates 180 degrees about the +z-axis. $\hat{Z}\ket{0} = \ket{0}$ and $\hat{Z} \ket{1} = -\ket{1}$

180 degree rotation about 45 degrees in the $xz$-plane $\hat{H}\ket{0} =\frac{\sqrt{ 2 }}{2}(\ket{0} + \ket{1}) = \ket{+}$ and $\hat{H} \ket{1} = \frac{\sqrt{ 2 }}{2}(\ket{0} - \ket{1}) = \ket{-}$

Rotates 90 degrees about the +z-axis. Square root of the $\hat{Z}$ gate, or the fourth root of the $\hat{I}$ gate. $\hat{S}\ket{0}=\ket{0}$ and $\hat{S}\ket{1} = i\ket{1}$

Rotates 45 degrees about the +z-axis. Square root of the $\hat{S}$ gate, or the fourth root of the $\hat{Z} gate$ $\hat{T}\ket{0} = \ket{0}$ and $\hat{T}\ket{1} = e^{i \frac{\pi}{4}}\ket{1}$