Quantum critical chains are well-described and understood by virtue of conformal field theory. Still, the meaning of the real space entanglement spectrum - the eigenvalues of the reduced density matrix - of such systems remains elusive in general, even when there is an additional quantum number available such as the spin or particle number. In this paper, we explore in detail the properties and structure of the reduced density matrix of critical XXZ spin-1/2 chains. We investigate the quantum/thermal correspondence between the reduced density matrix of a T = 0 pure quantum state and the thermal density matrix of an effective entanglement Hamiltonian. Using large scale DMRG and QMC simulations, we investigate the conformal structure of the spectra, the entanglement Hamiltonian, and temperature. We then introduce the notion of spin-resolved entanglement entropies, which display interesting scaling features.