We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual)
E
n
!
\mathcal {E}_n^!
of the Fomin–Kirillov algebras
E
n
\mathcal {E}_n
; these algebras are connected
N
\mathbb {N}
-graded and are defined for
n
≥
2
n \geq 2
. We establish that the algebra
E
n
!
\mathcal {E}_n^!
is module finite over its center (and thus satisfies a polynomial identity), is Noetherian, and has Gelfand–Kirillov dimension
⌊
n
/
2
⌋
\lfloor n/2 \rfloor
for each
n
≥
2
n \geq 2
. We also observe that
E
n
!
\mathcal {E}_n^!
is not prime for
n
≥
3
n \geq 3
. By a result of Roos,
E
n
\mathcal {E}_n
is not Koszul for
n
≥
3
n \geq 3
, so neither is
E
n
!
\mathcal {E}_n^!
for
n
≥
3
n \geq 3
. Nevertheless, we prove that
E
n
!
\mathcal {E}_n^!
is Artin–Schelter (AS-)regular if and only if
n
=
2
n=2
, and that
E
n
!
\mathcal {E}_n^!
is both AS-Gorenstein and AS-Cohen–Macaulay if and only if
n
=
2
,
3
n=2,3
. We also show that the depth of
E
n
!
\mathcal {E}_n^!
is
≤
1
\leq 1
for each
n
≥
2
n \geq 2
, conjecture that we have equality, and show that this claim holds for
n
=
2
,
3
n =2,3
. Several other directions for further examination of
E
n
!
\mathcal {E}_n^!
are suggested at the end of this article.