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The long-time dynamics of the d-dimensional spherical model with a non-conserved order parameter and quenched from an initial state with long-range correlations is studied through the exact calculation of the two-time autocorrelation and autoresponse functions. In the ageing regime, these are given in terms of non-trivial universal scaling functions of both time variables. At criticality, five distinct types of ageing are found, depending on the form of the initial correlations, while at low temperatures only a single type of ageing exists. The autocorrelation and autoresponse exponents are shown to be genetically different and to depend on the initial conditions. The scaling form of the two-time response functions agrees with a recent prediction coming from conformal invariance.