  We carry out an extensive investigation of conservation laws and potential
symmetries for the class of linear (1+1)-dimensional second-order parabolic
equations. The group classification of this class is revised by employing
admissible transformations, the notion of normalized classes of differential
equations and the adjoint variational principle. All possible potential
conservation laws are described completely. They are in fact exhausted by local
conservation laws. For any equation from the above class the characteristic
space of local conservation laws is isomorphic to the solution set of the
adjoint equation. Effective criteria for the existence of potential symmetries
are proposed. Their proofs involve a rather intricate interplay between
different representations of potential systems, the notion of a potential
equation associated with a tuple of characteristics, prolongation of the
equivalence group to the whole potential frame and application of multiple dual
Darboux transformations. Based on the tools developed, a preliminary analysis
of generalized potential symmetries is carried out and then applied to
substantiate our construction of potential systems. The simplest potential
symmetries of the linear heat equation, which are associated with single
conservation laws, are classified with respect to its point symmetry group.
Equations possessing infinite series of potential symmetry algebras are studied
in detail.
