  Quasiparticle interference (QPI) by means of scanning tunneling
microscopy/spectroscopy (STM/STS), angle resolved photoemission spectroscopy
(ARPES), and multi-orbital tight bind- ing calculations are used to investigate
the band structure and superconducting order parameter of LiFeAs. Using this
combination we identify intra- and interband scattering vectors between the
hole (h) and electron (e) bands in the QPI maps. Discrepancies in the band
dispersions inferred from previous ARPES and STM/STS are reconciled by
recognizing a difference in the $k_z$ sensitivity for the two probes. The
observation of both h-h and e-h scattering is exploited using phase-sensitive
scattering selection rules for Bogoliubov quasiparticles. From this we infer an
s$_\pm$ gap structure, where a sign change occurs in the superconducting order
parameter between the e and h bands.
