The structure of ordered expansions in powers of boson operators and of can
onical operators and the dual problem of operator reconstruction from order
ed moments is derived and applied to the complete Gaussian class of orderin
g. In particular, the interpolation lines between normal and antinormal ord
ering and between standard and antistandard ordering with Weyl symmetrical
ordering in their centre are dealt with in detail. The auxiliary operators
for expansions in symmetrical ordering are explicitly found in the Fock-sta
te representation and in other different representations. General and speci
alized formulae are derived for different ordering of powers of linear comb
inations of boson and of canonical operators which involve Hermite polynomi
als of operators. The link between symmetrical ordering of powers of boson
operators and of canonical operators is expressed by means of Jacobi polyno
mials. Some basic formulae of operator ordering and operator expansion are
collected for convenient use in the appendix.