Starting with a minimal model with the on-site Hilbert space reduced to only three effective valence centers CuO 7−,6−,5− 4 (nominally Cu 1+,2+,3+ ) we present an unified approach to the description of the variety of the local intra-unit-cell (IUC) order parameters determining a low-energy physics in cuprates. Central point of the model implies the occurrence of unconventional on-site quantum superpositions of the three valent states characterized by different hole occupation: n h =0,1,2 for Cu 1+,2+,3+ centers, respectively, different conventional spin: s=1/2 for Cu 2+ center and s=0 for Cu 1+,3+ centers, and different orbital symmetry:B1g for the ground states of the Cu 2+ center and A1g for the Cu 1+,3+ centers, respectively. The latter does result in a spontaneous orbital symmetry breaking accompanying the formation of the on-site mixed valence superpositions with emergence of the IUC orbital nematic order parameter of the B1g = B1g × A1g (∝ dx2−y2) symmetry. To describe the diagonal and off-diagonal, or quantum local charge order we develop an S=1 pseudospin model with a non-Heisenberg effective Hamiltonian that provides a physically clear description of "the myriad of phases" from a bare parent antiferromagnetic insulating phase to a Fermi liquid in overdoped cuprates. Conventional spin density ρs for mixed valence superpositions can vary inbetween 0 and 1 in accordance with the weight of the Cu 2+ center in the superposition. We show that the superconductivity and spin magnetism are nonsymbiotic phenomena with competing order parameters. Furthermore we argue that instead of a well-isolated Zhang-Rice (ZR) singlet 1 A1g the ground state of the hole Cu 3+ center in cuprates should be described by a complex 1 A1g-1,3 B2g-1,3 Eu multiplet, formed by a competition of conventional hybrid Cu 3d-O 2p b1g(σ) ∝ d x 2 −y 2 state and purely oxygen nonbonding O 2pπ states with a2g(π) and eux,y(π) symmetry. In contrast with inactive ZR singlet we arrive at several novel competing IUC orbital and spin-orbital order parameters, e.g., electric dipole and quadrupole moments, Ising-like net orbital magnetic moment, orbital toroidal moment, intra-plaquette's staggered order of Ising-like oxygen orbital magnetic moments. As a most impressive validation of the non-ZR model we explain fascinating results of recent neutron scattering measurements that revealed novel type of the IUC magnetic ordering in pseudogap phase of several hole-doped cuprates.