  We formulate a class of singular integral operators in arbitrarily many
parameters using mixed type characterizing conditions. We also prove a
multi-parameter representation theorem saying that a general operator in our
class can be represented as an average of sums of dyadic shifts, which implies
a new multi-parameter $T1$ theorem as a byproduct. Furthermore, an equivalence
result between ours and Journ\'e's class of multi-parameter operators is
established, whose proof requires the multi-parameter $T1$ theorem. These
results generalize to arbitrarily many parameters recent results of Hyt\"onen,
Martikainen, and Grau de la Herran.
