The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring
R
R
. Let
GradAlg
H
(
R
)
\operatorname {GradAlg}^H(R)
be the scheme parametrizing graded quotients of
R
R
with Hilbert function
H
H
. We prove there is a close relationship between the irreducible components of
GradAlg
H
(
R
)
\operatorname {GradAlg}^H(R)
, whose general member is a Gorenstein codimension
(
c
+
1
)
(c+1)
quotient, and the irreducible components of
GradAlg
H
′
(
R
)
\operatorname {GradAlg}^{H’}(R)
, whose general member
B
B
is a codimension
c
c
Cohen-Macaulay algebra of Hilbert function
H
′
H’
related to
H
H
. If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of
B
B
, this relationship actually determines a well-defined injective mapping from such “Cohen-Macaulay” components of
GradAlg
H
′
(
R
)
\operatorname {GradAlg}^{H’}(R)
to “Gorenstein” components of
GradAlg
H
(
R
)
\operatorname {GradAlg}^{H}(R)
, in which generically smooth components correspond. Moreover the dimension of the “Gorenstein” components is computed in terms of the dimension of the corresponding “Cohen-Macaulay” component and a sum of two invariants of
B
B
. Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.