DescriptionWe establish an equiconsistency between 1. a proper class of cardinals that are strong reflecting strongs exists2. a proper class of strong cardinals exists while all κ that are κ+2-strong have weakly indestructible κ+2-strength. Replacing κ+2 in (2) with λ makes no difference in consistency strength for sufficiently small λ. Indeed, this equiconsistency holds for λ well beyond the next measurable limit of measurables above κ, but λ must be below the next cardinal μ that is μ+2-strong. One direction of the equiconsistency of (1) and (2) is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed, and similar results are derived for supercompacts and supercompacts reflecting supercompacts.