The aim of this paper is to further explore an idea from J.-L. Loday and extend some of his results. We impose a natural and simple symmetry on a unit action over the most general quadratic relation which can be written. This leads us to two families of binary, quadratic and regular operads whose free objects are computed, as much as possible, as well as their duals in the sense of Ginzburg and Kapranov. Roughly speaking, free objects found here are in relation to rooted planar m-ary trees, triangular numbers and more generally m-tetrahedral numbers, homogeneous polynomials on $m$ commutative indeterminates over a field K and polygonal numbers. Involutive connected ℘-Hopf algebras are constructed.