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The main result of this paper is a construction of solutions to the reverseYang-Mills-Higgs flow converging in the $C^\infty$ topology to a criticalpoint. The construction uses only the complex gauge group action, which leadsto an algebraic classification of the isomorphism classes of points in theunstable set of a critical point in terms of a filtration of the underlyingHiggs bundle.  Analysing the compatibility of this filtration with theHarder-Narasimhan-Seshadri double filtration gives an algebraic criterion fortwo critical points to be connected by a flow line. As an application, we canuse this to construct Hecke modifications of Higgs bundles via theYang-Mills-Higgs flow. When the Higgs field is zero (corresponding to theYang-Mills flow), this criterion has a geometric interpretation in terms ofsecant varieties of the projectivisation of the underlying bundle inside theunstable manifold of a critical point, which gives a precise description ofbroken and unbroken flow lines connecting two critical points. For non-zeroHiggs field, at generic critical points the analogous interpretation involvesthe secant varieties of the spectral curve of the Higgs bundle.

The reverse Yang-Mills-Higgs flow in a neighbourhood of a critical point