A cooled spider for grazing-incidence collectors includes an outer ring, an inner ring and spokes that mechanically and fluidly connect the inner and outer rings. Cooling channels in the outer and inner rings and in the spokes define a general cooling-fluid flow path through the spider. The general cooling-fluid flow path has input and output points located substantially 180° apart so that the flow path diverges at the input point into two branch flow paths that flow in opposite directions through the spider, and then converge at the output point. Input and output cooling fluid manifolds are fluidly connected to the outer ring at the input and output points and serve to flow cooling fluid over the cooling-fluid flow path.