  In previous work by two of the present authors, twistors were re-interpreted
as 4-d spinors with a position dependence within the formalism of geometric
(Clifford) algebra. Here we extend that approach and justify the nature of the
position dependence. We deduce the spinor representation of the restricted
conformal group in geometric algebra, and use it to show that the position
dependence is the result of the action of the translation operator in the
conformal space on the 4-d spinor. We obtain the geometrical description of
twistors through the conformal geometric algebra, and derive the Robinson
congruence. This verifies our formalism. Furthermore, we show that this novel
approach brings considerable simplifications to the twistor formalism, and new
advantages. We map the twistor to the 6-d conformal space, and derive the
simplest geometrical description of the twistor as an observable of a
relativistic quantum system. The new 6-d twistor takes the r\^ole of the state
for that system. In our new interpretation of twistors as 4-d spinors, we
therefore only need to apply the machinery already known from quantum mechanics
in the geometric algebra formalism, in order to recover the physical and
geometrical properties of 1-valence twistors.
