We consider the Hardy-Hénon parabolic equation ∂tu=Δu+|x|−γ|u|αu, t>0, x∈RN∖{0}, N≥1, 0<γ<min(2,N) and α>0. The existence of solutions in C([0,T],Lq(RN)), 1<q<∞, q≥qc:=Nα/(2−γ) with initial data u0∈Lq(RN) has been established recently. It is also known that uniqueness holds in C([0,T],Lq(RN)) for q>qc, q>Qc:=N(α+1)/(N−γ), while for the other values of q it holds under additional restrictions. In this paper, we consider the problem of uniqueness for the limiting case q=max(qc,Qc). Our objective is to see whether or not we can remove the additional restrictions. Here two different situations arise:–A single critical case: when q=qc>Qc and u0∈Lq(RN) or q=Qc>qc and u0 in the Lorentz space Lq,α+1(RN), we prove that uniqueness holds in C([0,T],Lq(RN)).–The double-critical case: when q=qc=Qc, that is N≥3, 0<γ<2, α=(2−γ)/(N−2) and q=N/(N−2). We prove non-uniqueness in C([0,T],LNN−2(RN)) for any initial value u0∈LNN−2(RN). We also give a uniqueness criterion for the double-critical case and we show that it is optimal in the sense that it characterizes the largest space in which uniqueness holds. The uniqueness criterion as well as the uniqueness results involve Lorentz spaces and show that the second index in Lorentz spaces plays a crucial role. The proofs rely on some estimates for the heat kernel in Lorentz spaces and use the singular solutions known for the stationary Dirichlet problem on the unit ball.