The nature of polyamorphism and amorphous-to-amorphous transition is investigated by means of an exactly solvable model with quenched disorder, the spherical s+p multi-spin interaction model. The analysis is carried out in the framework of Replica Symmetry Breaking theory and leads to the identification of low temperature glass phases of different kinds. Besides the usual 'one-step' solution, known to reproduce all basic properties of structural glasses, also a physically consistent 'two-step' solution arises. More complicated phases are found as well, as temperature is further decreased, expressing a complex variety of metastable states structures for amorphous systems.PACS numbers: 75.10. Nr,64.70.Pf,71.55.Jv In recent years increasing evidence has been collected for the existence of amorphous to amorphous transitions (AAT), in various glass-forming substances as, e.g., vitreous Germania and Silica, where the coordination changes abruptly under pressure shifts [1]. One refers to this phenomenon as "polyamorphism" [1,2]. Like the liquid-glass transition also the AAT is not a thermodynamic phase transition, but it amounts to a qualitative change in the relaxation dynamics, apparently expressing a recombination of the glass structure. Other kinds of AAT are known to occur, e.g., in porous silicon [3], in undercooled water [4,5], in copolymer micellar systems [6] and in polycarbonate and polystyrene glassy polymers [7].A number of theoretical models has been introduced to describe systems undergoing AAT, as, e.g., a model of hard-core repulsive colloidal particles subject to a shortrange attractive potential [8,9,10]. Another instance is the spherical p-spin model on lattice gas of Ref. [11]. In this paper we consider a model with multibody quenched disordered interactions where various AAT's occur as well: the spherical s + p-spin model.Our analytic investigation is performed by applying Parisi's Replica Symmetry Breaking (RSB) theory [12]. In the framework of the theoretical description of glasses and, more generally, of disordered systems, RSB theory provides, in a broad variety of instances, a rather deep and complex mean-field insight. RSB solutions so far encountered, representing physically stable phases, are either one step RSB (mean-field glass) or implement a continuous hierarchy of breakings (mean-field spin-glass). One step RSB (1RSB) solutions are, e.g., found in the Ising p-spin [13,14] and Potts [15] models with quenched disorder in a low temperature interval [43] or in the spherical p-spin model below the static critical temperature [18], else in optimization problems mapped into dilute spin-glass systems such as the XOR-SAT (where it correctly describes the whole UNSAT phase) [19], or the K-SAT, with K > 2 [20], in a certain interval of connectivity values next to the SAT/UNSAT transition [21]. The continuous, or full (FRSB) solution describes, instead, the low temperature phase of the mean-field version of the Ising spin-glass, i.e., the Sherrington-Kirkpatrick model [16]. Low temperature phas...