Question:  How is a photon its own antiparticle?
As I understand it, the fundamental difference between matter
and antimatter is the spin of the particle....
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Answer 1:  Fundamental difference between particle and antiparticle
is NOT a difference in spin.  An antiparticle can be thought
of as a time reversed particle.  If you start with description
of a charged particle and time reverse, you get a particle
with opposite charge and spin ( i mean that you get description
of a particle with opposite charge and spin ).  But if you start
with a charge neutral particle like photon, you get the same particle
with opposite spin.  So we say that photon is its own antiparticle.
This follows only because photon does not have any kind of charge.
jasjeet s bagla

Answer 2: might note that neutrinos also have no charge, and it
is not at the moment certain whether an antineutrino
is the same as a neutrino or not.  If the neutrino is
its own antiparticle, then that means the neutrino must
have a mass (the argument is in fact based on spin, which
might be where you heard that the difference between
a particle and antiparticle is in the spin).
Arthur Smith

Answer 3:
Neutrinoes do not have electric charge but they have
the Leptonic charge.  I used the word charge in its
generalised sense.  I
Sorry for not making this explicit.
jasjeet s bagla