A well-known elegant geometric approach to Bézier and B-spline curves is based on the so-called blossoming principle: associating with a polynomial curve or piecewise polynomial curve an appropriate normal curve in a higher-dimensional space, control points and auxiliary points occurring in important algorithms are obtained as blossoms, which are intersections of appropriate osculating flats of the normal curve. Extensions to geometrically continuous polynomial splines and to Chebyshev splines are known. In the present paper, the author extends the geometric approach to blossoming to a very general class of splines, whose segments may be taken from different Chebyshev spaces and are tied together by geometric continuity conditions. It is shown that all essential properties of control point based curve design hold in this case as well. Moreover, a thorough investigation of an associated subblossoming principle is given and applied to simplify the general blossoming theory.