A valve stem packing assembly can seal a valve stem to a valve body having a body cavity. The packing assembly includes a packing ring circumscribing the valve stem within a stem opening extending axially through the valve body. A primary dynamic seal is positioned to seal a dynamic leak path between the packing ring and the valve stem. A secondary dynamic seal is spaced axially apart and functionally independent from the primary dynamic seal and positioned to redundantly seal the dynamic leak path. A primary static seal is positioned to seal a static leak path between the packing ring and the valve body. A secondary static seal is spaced axially apart and functionally independent from the primary static seal and positioned to redundantly seal the static leak path.