  We study the percolation of randomly rotating patchy particles on $11$
Archimedean lattices in two dimensions. Each vertex of the lattice is occupied
by a particle, and in each model the patch size and number are monodisperse.
When there are more than one patches on the surface of a particle, they are
symmetrically decorated. As the proportion $\chi$ of the particle surface
covered by the patches increases, the clusters connected by the patches grow
and the system percolates at the threshold $\chi_c$. We combine Monte Carlo
simulations and the critical polynomial method to give precise estimates of
$\chi_c$ for disks with one to six patches and spheres with one to two patches
on the $11$ lattices. For one-patch particles, we find that the order of
$\chi_c$ values for particles on different lattices is the same as that of
threshold values $p_c$ for site percolation on same lattices, which implies
that $\chi_c$ for one-patch particles mainly depends on the geometry of
lattices. For particles with more patches, symmetry become very important in
determining $\chi_c$. With the estimates of $\chi_c$ for disks with one to six
patches, by analyses related to symmetry, we are able to give precise values of
$\chi_c$ for disks with an arbitrary number of patches on all $11$ lattices.
The following rules are found for patchy disks on each of these lattices: (i)
as the number of patches $n$ increases, values of $\chi_c$ repeat in a periodic
way, with the period $n_0$ determined by the symmetry of the lattice; (ii) when
$\mod(n,n_0)=0$, the minimum threshold value $\chi_{\rm min}$ appears, and the
model is equivalent to site percolation with $\chi_{\rm min}=p_c$; (iii) disks
with $\mod(n,n_0)=m$ and $n_0-m$ ($m<n_0/2$) share the same $\chi_c$ value.
