There is some confusion surrounding the pruning of hydrangeas, because there are several different species that are excellent for the garden, but not all are pruned at the same time. Pruning at the wrong time can abort flowering; each species should be pruned in relation to when it forms its flower buds.
Prune smooth hydrangea (H. arborescens) and panicle, or Pee Gee, hydrangea (H. paniculata) in late winter or early spring. These species set their flower buds on their new growth.
Prune bigleaf (H. macrophylla), oakleaf (H. quercifolia) and climbing (H. anomala subsp. petiolaris) hydrangeas just after they flower. These species form their flower buds in the summer and fall, or "on the old wood." Therefore pruning them in the winter or spring would remove the buds before they have a chance to bloom, and you'd see no flowers that year.
A very cold winter will sometimes kill bigleaf hydrangea buds—keep that in mind if you pruned them at the right time but still have no flowers. Read about changing their flower color.
Images courtesy of the Missouri Botanical Garden PlantFinder
Below left: H. arborescens Below right: H. paniculata 'Tardiva'
Left: H. macrophylla 'Nikko Blue'
Left: H. quercifolia Snow Queen
Above: H. anomala subsp. petiolaris
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