  Jones proposed the study of two subfactors of a $II_1$ factor as a
quantization of two closed subspaces in a Hilbert space. The Pimsner-Popa
probabilistic constant, Sano-Watatani angle, interior and exterior angle, and
Connes-St{\o}rmer relative entropy (along with a slight variant of it) are a
few key invariants for pair of subfactors that analyze their relative position.
In practice, however, the explicit computation of these invariants is often
difficult. In this article, we provide an in-depth analysis of a special class
of two subfactors, namely a pair of spin model subfactors of the hyperfinite
type $II_1$ factor $R$. We first characterize when two distinct $n\times n$
complex Hadamard matrices give rise to distinct spin model subfactors. Then, a
detailed investigation has been carried out for pairs of (Hadamard equivalent)
complex Hadamard matrices of order $2\times 2$ as well as Hadamard inequivalent
complex Hadamard matrices of order $4\times 4$. To the best of our knowledge,
this article is the first instance in the literature where the exact value of
the Pimsner-Popa probabilistic constant and the noncommutative relative entropy
for pairs of (non-trivial) subfactors have been obtained. Furthermore, we prove
the factoriality of the intersection of the corresponding pair of subfactors
using the `commuting square technique'. En route, we construct an infinite
family of potentially new subfactors of $R$. All these subfactors are
irreducible with Jones index $4n,n\geq 2$. As a consequence, the rigidity of
the interior angle between the spin model subfactors is established. Last but
not least, we explicitly compute the Sano-Watatani angle between the spin model
subfactors.
