  The competing orders in the particle-particle (P-P) channel and the
particle-hole (P-H) channel have been proposed separately to explain the
pseudogap physics in cuprates. By solving the Bogoliubov-deGennes equation
self-consistently, we show that there is a general complementary connection
between the d-wave checkerboard order (DWCB) in the particle-hole (P-H) channel
and the pair density wave order (PDW) in the particle-particle (P-P) channel. A
small pair density localization generates DWCB and PDW orders simultaneously.
The result suggests that suppressing superconductivity locally or globally
through phase fluctuation should induce both orders in underdoped cuprates. The
presence of both DWCB and PDW orders with $4a \times 4a$ periodicity can
explain the checkerboard modulation observed in FT-STS from STM and the
puzzling dichotomy between the nodal and antinodal regions as well as the
characteristic features such as non-dispersive Fermi arc in the pseudogap
state.
