  We call a pair of distinct prime powers $(q_1,q_2) = (p_1^{a_1},p_2^{a_2})$ a
Hasse pair if $|\sqrt{q_1}-\sqrt{q_2}| \leq 1$. For such pairs, we study the
relation between the set $\mathcal{E}_1$ of isomorphism classes of elliptic
curves defined over $\mathbb{F}_{q_1}$ with $q_2$ points, and the set
$\mathcal{E}_2$ of isomorphism classes of elliptic curves over
$\mathbb{F}_{q_2}$ with $q_1$ points. When both families $\mathcal{E}_i$
contain only ordinary elliptic curves, we prove that their isogeny graphs are
isomorphic. When supersingular curves are involved, we describe which curves
might belong to these sets. We also show that if both the $q_i$'s are odd and
$\mathcal{E}_1 \cup \mathcal{E}_2 \neq \emptyset$, then $\mathcal{E}_1 \cup
\mathcal{E}_2$ always contains an ordinary elliptic curve. Conversely, if $q_1$
is even, then $\mathcal{E}_1 \cup \mathcal{E}_2$ may contain only supersingular
curves precisely when $q_2$ is a given power of a Fermat or a Mersenne prime.
In the case of odd Hasse pairs, we could not rule out the possibility of an
empty union $\mathcal{E}_1 \cup \mathcal{E}_2$, but we give necessary
conditions for such a case to exist. In an appendix, Moree and Sofos consider
how frequently Hasse pairs occur using analytic number theory, making a
connection with Andrica's conjecture on the difference between consecutive
primes.
