We consider the continuum scaling limit of the infinite series of Yang–Baxter integrable logarithmic minimal models LM(p,p′) as ‘rational’ logarithmic conformal field theories with extended W symmetry. The representation content is found to consist of 6pp′−2p−2p′W-indecomposable representations of which 2p+2p′−2 are of rank 1, 4pp′−2p−2p′ are of rank 2, while the remaining 2(p−1)(p′−1) are of rank 3. We identify these representations with suitable limits of Yang–Baxter integrable boundary conditions on the lattice. The W-indecomposable rank-1 representations are all W-irreducible while we present a conjecture for the embedding patterns of the W-indecomposable rank-2 and -3 representations. The associated W-extended characters are all given explicitly and decompose as finite non-negative sums of W-irreducible characters. The latter correspond to W-irreducible subfactors and we find that there are 2pp′+(p−1)(p′−1)/2 of them. We present fermionic character expressions for some of the rank-2 and all of the rank-3 W-indecomposable representations. To distinguish between inequivalent W-indecomposable representations of identical characters, we introduce ‘refined’ characters carrying information also about the Jordan-cell content of a representation. Using a lattice implementation of fusion on a strip, we study the fusion rules for the W-indecomposable representations and find that they generate a closed fusion algebra, albeit one without identity for p>1. We present the complete set of fusion rules and interpret the closure of this fusion algebra as confirmation of the proposed extended symmetry. Finally, 2pp′ of the W-indecomposable representations are in fact W-projective representations and they generate a closed fusion subalgebra.