  The standard operational probabilistic framework (within which we can
formulate Operational Quantum Theory) is time asymmetric. This is clear because
the conditions on allowed operations are time asymmetric. It is odd, though,
because Schoedinger's equation is time symmetric and probability theory does
not care about time direction. In this work we provide a time symmetric
framework for operational theories in general and for Quantum Theory in
particular.
  The clearest expression of the time asymmetry of standard Operational Quantum
Theory is that the deterministic effect is unique - meaning there is only one
way to ignore the future - while deterministic (i.e normalised) states are not
unique. In this paper, this time asymmetry is traced back to a time asymmetric
understanding of the most basic elements of an operational theory - namely the
operations (or boxes) out of which circuits are built. We modify this allowing
operations to have classical incomes as well as classical outcomes on these
operations. We establish a time symmetric operational framework for circuits
built out of operations. In particular, we demand that the probability
associated with a circuit is the same whether we calculate it forwards in time
or backwards in time. We do this by imposing various double properties. These
are properties wherein a forward in time and a backward in time version of the
same property are required. In this paper we provide a new causality condition
which we call double causality.
