What is the inverse of Energy? Is there a name for the inverse of Energy $1/E$ or the inverse of the Joule $1/Joule$? It comes up if you divide the mass-energy relation by energy (so you get a constant of 1).
 A: Short answer: Time.
Long answer:
The conjugate quanta of energy is time, one can look at energy as the generalized momenta associated with time.
With that is mind one can go to QM essentials and proclaim:
$$
\Delta E \Delta t \geq \frac{\hbar}{2}
$$
As a consequence, one can say:
$$
\frac{1}{E}\propto t
$$
Edit:
Even longer answer:
I agree with gented below, that this answer can be misleading and promote a wrong view of physics, hence this edit.
From the viewpoint of Noether theorem, and its meaning for field theory, one can look at transformations as operators. Every linear transformation that is continuously connected to the identity transformation can be written down as:
$$
F(\vec{x}+\vec{dx})=F+dF,
$$
with $dF=\sum\frac{dF}{dx_i}dx_i$. If we further assign a deformation parameter $\alpha$ such that the change in function is due to the deformation parameter alone we can write:
$$
dF=\sum \frac{dF}{dx_i}\frac{dx_i}{d\alpha}d\alpha =d\alpha \left(\sum\frac{dx_i}{d\alpha}\frac{d}{dx_i}\right)F
$$
The quantity $\left(\sum\frac{dx_i}{d\alpha}\frac{d}{dx_i}\right)$ is called the generator of the transformation such that the transformation can be written as an exponent like so:
$$
T_a=e^{Ga}.
$$
Notice this satisfies $T_0=I$. There are several ways to find the generator(s) of a transformation, but it so happens that the generators of spatial translations is momentum such that if you want the transformation $x\rightarrow x+a$, it is given by $T_a=e^{\frac{i\hat{P}a}{\hbar}}$ where $\hat{P}$ is given by
$$
\hat{P}= -i\hbar \frac{\partial}{\partial x}.
$$ 
It is in this sense that we can identify reciprocal physical quantities, the generator of $x$ translation is momentum, so in a sense momentum is the reciprocal of location. It turns out that the generator of time translations is the Hamiltonian, as the time translation operator can be written as 
$$
U(t,t+\Delta t)=e^{\frac{-i}{\hbar}\int_t^{t+\Delta t}\mathcal{H}dt},
$$
Where the Hamiltonian represents the Energy of the system, and is promoted to an operator status.
Under some conditions this can be simplified further to the following:
$$
U=e^{\frac{-i}{\hbar}\mathcal{H}\Delta t},
$$
so in this sense the energy is the 'reciprocal' of time.
There are volumes of information and infrastructure missing from this answers, but in a nutshell this was all that I could manage. 
This 'reciprocity' can and is used say in elementary particle physics as a 'rule of thumb' for particle decay times, where the more massive the particle the faster it should decay. Although this is just a guesstimate technique and more often than not doesn't really work.
