  We study the symmetry and the spatial uniformity of orientational order of
the biaxial nematic phase in the light of recent experimental observations of
phase biaxiality in thermotropic bent-core and calamitic-tetramer nematics. We
present evidence supporting monoclinic symmetry, instead of the usually assumed
orthorhombic. We describe the use of deuterium NMR to differentiate between the
possible symmetries. We present the spatial aspects of biaxial order in the
context of the cluster model, wherein macroscopic biaxiality can result from
the field-induced alignment of biaxial and possibly polar domains. We discuss
the implications of different symmetries, in conjunction with the microdomain
structure of the biaxial phase, on the alignment of biaxial nematics and on the
measurements of biaxial order.
