  Via a series of Kaluza-Klein (KK) and Scherk-Schwarz (SS) compactifications
we relate BPS attractors and their complete (in general non-BPS) flows to a
Minkowski vacuum in gauged supergravities with vanishing scalar potential in 4,
5, and 6 dimensions. This way we can look at a class of extremal non-BPS black
holes and strings from IIB string theory viewpoint, keeping 4 supercharges on
the horizon. Our results imply the existence of a dual 2d N = (0,2)
superconformal field theory (SCFT) that originates from a parent N=(4,4) theory
living on a D1-D5 system.
  This is achieved starting from the BPS black string in 6d with an AdS_3xS^3
attractor and taking two different routes to arrive at a 1/2 BPS AdS_2xS^2
attractor of a non-BPS black hole in 4d N=2 flat gauged supergravity. The two
inequivalent routes interchange the order of KK reduction on AdS_3 and SS
reduction on S^3. We also find the commutator between the two operations after
performing a duality transformation: on the level of the theory the result is
the exchange of electric with magnetic gaugings; on the level of the solution
we find a flip of the quartic invariant I_4 to -I_4.
