A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in classical many-body systems after a quench from an initially disordered system to a temperature $T\le T_c$, at or below the critical temperature $T_c$. Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents ${\lambda }_C={\lambda }_R$, the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments $y=t/s\gg 1$ to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent Θ is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures $T\le T_c$. Relevance to experiments is also discussed.