Here we initiate an investigation into the classeOof extended Ockham algebras, namely bounded distributive lattices endowed with a commuting pair of unary operations, one of which is an endomorphism and the other is a dual endomorphism. A consideration of the subdirectly irreducible algebras leads, via Priestley duality, to a description of all finite subdirectly irreducibleeO-algebras. In particular, we show that there are precisely nine non-isomorphic subdirectly irreducible symmetric extended de Morgan algebras, all of which are simple. We also describe the corresponding lattice of varieties.