  The aim of this paper is to further explore an idea from J.-L. Loday briefly
exposed in [5]. 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, as well
as their duals in the sense of Ginzburg and Kapranov are computed. Roughly
speaking, free objects found here are in relation to $m$-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 P-Hopf algebras are constructed and a link to genomics is
discussed. We also propose in conclusion some open questions.
