In addition to Peter's nice collection of self-dualities there are the Kleisli and Eilenberg-Moore categories of "the" covariant power-set monad (there are really two such monads but either will do), respectively Rel and complete semilattices, both self-dual. One that Mike Barr introduced me to is the subcategory of Rel whose morphisms are the partial injections, those binary relations such that if (x,y) and (x,z) are both present then y = z and likewise for their converses. My personal favorites are finite chains with bottom (showing that \Delta, as the base category of the presheaf category of simplicial sets, comes very close to being self-dual; had \Delta itself been self-dual, Set^{\Delta\op} and Set^\Delta would have been the same thing), and semilattices with a top and all nonempty sups (my candidate for a self-dual system of event/state structures before I replaced it with Chu spaces). One should also mention the topological vector spaces in Barr's book on *-autonomous categories, whose self-duality does for the finite-dimensional vector spaces mentioned by Peter what Pontryagin duality does for finite abelian groups. A feature of Chu spaces I particularly like is that each of the above, as well as finite-dimensional vector spaces and finite abelian groups, can be described as that full subcategory of Chu(Set,K) (K = 2 except for vector spaces and abelian groups) consisting of biextensional Chu spaces whose rows and columns satisfy the same closure conditions. For example the category of finite-dimensional vector spaces over GF(2) (a sneaky way to stick to Chu(Set,2)) embeds in Chu(Set,2) as precisely those finite biextensional Chu spaces whose rows and columns, viewed as bit vectors in the sense a machine-language programmer understands the concept, are both closed under bitwise XOR. This example is given as an exercise at the end of Chapter 2 of http://boole.stanford.edu/pub/coimbra.pdf, my notes for the July 1999 Coimbra School cotaught with John Baez and Cristina Pedicchio. Proposition 2.2 in the same chapter obtains complete semilattices (of any cardinality) as those Chu spaces whose rows and columns are closed under bitwise OR, with the self-duality of CSLat as the immediate Corollary 2.3. This shows that the self-dualities CSLat and Vct_{GF(2)}, at least for finite objects, arise identically except for how they combine their bit-vectors, namely with respectively OR and XOR. Vaughan Peter Freyd wrote:
On the subject of favorite dualities:
Surely the most important are the self-dualities and the most important of these (so important we stop noticing it as we age) is the category of finite-dimensional vector spaces over a given field.
Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules. (For 50 years I've been trying to turn this into an exercise in abelian categories. There's a nice reduction down to the proposition that R/Z is a cogenerator for the category of compact abelian groups, but that fact seems to require some non-trivial functional analysis.) Strange that two of the most important "dualities" are both Pontryagin's. The other is in algebraic topology theory.
Then, of course there's my present favorite: the category of finitely presented group-valued functors from the category of finitely presented modules over a commutative ring.