Auslander and Bridger [Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969] introduced the notions of
n
n
-spherical modules and
n
n
-torsionfree modules. In this paper, we construct an equivalence between the stable category of
n
n
-spherical modules and the category of modules of grade at least
n
n
, and provide its Gorenstein analogue. As an application, we prove that if
R
R
is a Gorenstein local ring of Krull dimension
d
>
0
d>0
, then there exists a stable equivalence between the category of
(
d
−
1
)
(d-1)
-torsionfree
R
R
-modules and the category of
d
d
-spherical modules relative to the local cohomology functor.