  We define a partial order $\mathcal{P}_n$ on permutations of any given size
$n$, which is the image of a natural partial order on inversion sequences. We
call this the ``middle order''. We demonstrate that the poset $\mathcal{P}_n$
refines the weak order on permutations and admits the Bruhat order as a
refinement, justifying the terminology. These middle orders are distributive
lattices and we establish some of their combinatorial properties, including
characterization and enumeration of intervals and boolean intervals (in
general, or of any given rank), and a combinatorial interpretation of their
Euler characteristic. We further study the (not so well-behaved) restriction of
this poset to involutions, obtaining a simple formula for the M\"obius function
of principal order ideals there. Finally, we offer further directions of
research, initiating the study of the canonical Heyting algebra associated with
$\mathcal{P}_n$, and defining a parking function analogue of $\mathcal{P}_n$.
