  The Cauchy problem for the Hardy-H\'enon parabolic equation is studied in the
critical and subcritical regime in weighted Lebesgue spaces on the Euclidean
space $\mathbb{R}^d$. Well-posedness for singular initial data and existence of
non-radial forward self-similar solution of the problem are previously shown
only for the Hardy and Fujita cases ($\gamma\le 0$) in earlier works. The
weighted spaces enable us to treat the potential $|x|^{\gamma}$ as an increase
or decrease of the weight, thereby we can prove well-posedness to the problem
for all $\gamma$ with $-\min\{2,d\}<\gamma$ including the H\'enon case
($\gamma>0$). As a byproduct of the well-posedness, the self-similar solutions
to the problem are also constructed for all $\gamma$ without restrictions. A
non-existence result of local solution for supercritical data is also shown.
Therefore our critical exponent $s_c$ turns out to be optimal in regards to the
solvability.
