Let
G
=
S
p
i
n
[
4
n
+
1
]
\mathbb {G}=\mathrm {Spin}[4n+1]
be the connected, simply connected complex Lie group of type
B
2
n
B_{2n}
and let
G
=
S
p
i
n
(
p
,
q
)
G=\mathrm {Spin}(p,q)
(
p
+
q
=
4
n
+
1
)
(p+q=4n+1)
denote a (connected) real form. If
q
∉
{
0
,
1
}
q \notin \left \{0,1\right \}
,
G
G
has a nontrivial fundamental group and we denote the corresponding nonalgebraic double cover by
G
~
=
S
p
i
n
~
(
p
,
q
)
\tilde {G}=\widetilde {\mathrm {Spin}}(p,q)
. The main purpose of this paper is to describe a symmetry in the set of genuine parameters for the various
G
~
\tilde {G}
at certain half-integral infinitesimal characters. This symmetry is used to establish a duality of the corresponding generalized Hecke modules and ultimately results in a character multiplicity duality for the genuine characters of
G
~
\tilde {G}
.