Are capacitors and inductors mirror images of each other?

Capacitors and inductors are images of one another under the self-inverse mapping that transforms a linear electrical network to its dual network.

The network duality transformation maps the network's graph to its topological dual graph,then all the impedances (either as lone-frequency complex scalars or as Laplace transfer functions) in the dual graph links become their reciprocals and current sources become voltage sources and contrariwise.

The physical meaning of this mapping is that we are finding a network where the voltages and currents in the network's state equations swap roles. So this is the reason for your observation: your two equations result from one another if you swap the roles of the voltage and current.

Some common examples: the Norton equivalent source is the dual of the Thévenin equivalent source and contrariwise. Likewise the star-delta transformation is an evocative example, showing how loops become links and contrariwise in the dual graph.

For a slightly more prosaic elucidation, again using the electrical circuit paradigm, I find it helpful to rework everything into a (rather approximately, surely) "pseudo-Ohm's Law" or "impedance-oriented" configuration:

Inductance: $$V=\frac{\textrm{d}I}{\textrm{d}t}\cdot L$$

Resistance: $$V = I\cdot R$$

Capacitance: \begin{align}\Delta V& = \Delta q\cdot C \\&= \int_{\Delta t}{I\!\left(t'\right)\mathrm{d}t'}\cdot C\;.\end{align}

So, yes: inductance and capacitance mirror each other in that the former relates the voltage to the derivative of the current, whereas the latter relates the voltage to the integral of the current.