  A less studied numerical characteristic of periodic orbits of area preserving
twist maps of the annulus is the twist or torsion number, called initially the
amount of rotation. It measures the average rotation of tangent vectors under
the action of the derivative of the map along that orbit, and characterizes the
degree of complexity of the dynamics. The aim of this paper is to give new
insights into the definition and properties of the twist number, and to relate
its range to the order properties of periodic orbits. We derive an algorithm to
deduce the exact value or a demi--unit interval containing the exact value of
the twist number. We prove that at a period-doubling bifurcation threshold of a
mini-maximizing periodic orbit, the new born doubly periodic orbit has the
absolute twist number larger than the absolute twist of the original orbit
after bifurcation. We give examples of periodic orbits having large absolute
twist number, that are badly ordered, and illustrate how characterization of
these orbits only by their residue can lead to incorrect results. In connection
to the study of the twist number of periodic orbits of standard--like maps we
introduce a new tool, called 1-cone function. We prove that the location of
minima of this function with respect to the vertical symmetry lines of a
standard-like map encodes a valuable information on the symmetric periodic
orbits and their twist number.
