We report some unusual phase behaviour, of general implication for condensed matter, on the polymer poly-4-methyl pentene-1 (P4MP1) induced by changes in pressure (P) and temperature (T), as observed by in-situ X-ray diffraction and high pressure DSC. Upon increasing pressure beyond a threshold value, the polymer, crystalline at ambient conditions, looses its crystalline order isothermally. The process is reversible. This behaviour is observed in two widely separated temperature regions, one below the glass transition temperature (< 50°C) and one close to the melting temperature (250°C), thus showing solid state amorphization and inversion in the melting temperature with increasing pressure. This further suggests inverse melting, i.e. re-entrant of the two widely separated liquid and amorphous phases along the T-axis at fixed P. This is confirmed experimentally as disordering in the crystalline structure on cooling. The inverse melting in P4MP1 raises the possibility of exothermic melting and endothermic crystallization as anticipated by Tammann (1903). The anticipated exothermic melting and endothermic crystallization is confirmed experimentally in the one component system P4MP1. We are observing similar features in a range of polymers.