You can consider X and Z gates as "inversion" in computational basis and circular and Hadamard bases, respectively.

Lets start with X. It holds that
$$
X|0\rangle = |1\rangle\,\,\,\,\,\,\ X|1\rangle = |0\rangle,
$$
so X is analogical to classical negation, i.e. it converts 0 to 1 and conversely.

Instead of computational basis $\{|0\rangle, |1\rangle\}$, you can express qubits as combination of members of Hadamard basis $\{|+\rangle, |-\rangle\}$, where
$$
|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)
\\
|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)
$$

You can verify that
$$
Z|+\rangle = |-\rangle\,\,\,\,\,\,\ Z|-\rangle = |+\rangle,
$$

Circular basis is composed of $\{|\uparrow\rangle, |\downarrow\rangle\}$ (*note that I was not able to find proper symbol for circular arrows*), where
$$
|\uparrow\rangle = \frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)
\\
|\downarrow\rangle = \frac{1}{\sqrt{2}}(|0\rangle-i|1\rangle)
$$

You can again verify that
$$
Z|\uparrow\rangle = |\downarrow\rangle\,\,\,\,\,\,\ Z|\downarrow\rangle = |\uparrow\rangle,
$$

All Pauli gates also define rotation around axes x, y and z. Consider $A \in \{X,Y,Z\}$ then rotation by angle $\theta$ around axis $a \in \{x,y,z\}$ is defined as
$$
R_a(\theta) = \mathrm{e}^{-i\frac{\theta}{2}A}
$$
Note that the exponential is so-called matrix exponential.