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We introduce a class of relativistic fluid states satisfying the relativistic local thermodynamical equilibrium postulate (abbreviated as relativistic (LTE) postulate). States satisfying this postulate, are states "near equilibrium" (a term defined precisely in the course of the paper) and permit us to attach a fictitious "local thermodynamical equilibrium" state that fits event by event the actual fluid state. They single out an admissible class of rest frames relative to which thermodynamical variables like the energy density, thermodynamical pressure, stresses, particle number density (or densities) measured by observers at rest relative to these frames are becoming frame independent provided second (or higher) order deviations from the fictitious state of "local thermodynamical equilibrium" are ignored. We have verified this property for a large class of theories of relativistic dissipation that include the Hiscock-Lindblom class of first order theories, the Eckart and Landau-Lifshitz theories, the Israel-Stewart transient thermodynamics, the Liu-Müller-Ruggeri theory, fluids of divergence type and the latest developed (BDNK) theory. Moreover, the phenomenological equations describing first order deviations from the fictitious "local thermodynamical equilibrium" state satisfy equations that remain form invariant under change of frame within the class of admissible frames. We proved this property for the Hiscock-Lindblom class of first order theories the Eckart and Landau-Lifshitz theories, the Israel-Stewart transient thermodynamics and the Liu-Müller-Ruggeri theory of relativistic dissipation the (BDNK) theory and we expect that the same property to hold for the class of relativistic fluids of divergence type.