The superficial similarity between the Chu construction and the Hyland-Tan
double glueing construction **G** has been observed widely. This paper establishes a
more formal mathematical relationship between the two.

We show that double glueing on relations subsumes the Chu construction on sets:
we present a full monoidal embedding of the category **chu**(Set,K) of biextensional
Chu spaces over K into **G**(Rel^K), and a full monoidal embedding of
the category **Chu**(Set,K) of Chu spaces over K into **IG**(Rel^K), where
we define **IG**, the *intensional double glueing construction*,
by substituting multisets for sets in **G**.

We define a biextensional collapse from **IG** to **G** which extends
the familiar notion on Chu spaces. This yields a new interpretation of the
monic specialisation implicit in **G** as a form of biextensionality.