Modern determinism in the scientific sense is basically only about mathematical prediction--if the universe is deterministic, that means complete knowledge of the universe's physical state at an earlier time can be used to predict the physical state at a later time with perfect accuracy, no additional notions associated with philosophical ideas about "causation" are included.
Here is philosopher of science Timothy Maudlin talking about the subject:
As for causation, everyday causal locutions are highly context-sensitive and subject to pragmatic considerations. One does not want any foundational physical concepts to have these features, so at least everyday causal locutions cannot be translated cleanly into basic physical terms. Furthermore, physics gets on fine without mention of causation: dynamical law does all the work. So there is no need to admit some new irreducible notion of causation to make sense of physics. A deterministic physics might endorse a claim like “earlier global physical states cause later global physical states”, but that claim is of little use for everyday talk of causes.
Also note that because all the known fundamental laws of physics have the property of either time-symmetry or CPT-symmetry, it is equally possible to predict earlier states given complete knowledge of the universe's state at a later time. The various arrows of time we see (asymmetries between past and future) are thought by most physicists to all be due to the the universe being in a low-entropy state around the time of the Big Bang, for reasons that are not really understood. There is a good discussion of the arrow of time and its connection to low-entropy initial conditions in Brian Greene's book The Fabric of the Cosmos.
So if the philosophical notion of "causality" includes the notion that past events cause future ones but not vice versa, there is really nothing corresponding to this idea in modern physics.
This section of the SEP article on causality also mentions that Bertrand Russell endorsed a similar sort of eliminativism about causation in his 1913 paper "On the Notion of Cause" (included in The Collected Papers of Bertrand Russell, Volume 6 and also available online here). In that paper, Russell commented that "The law of causality, I believe, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm", and that "In the motions of mutually gravitating bodies, there is nothing that can be called a cause, and nothing that can be called an effect; there is merely a formula."
Russell also commented on the issue of the time-symmetry of the laws of physics, saying:
The law makes no difference between past and future: the future "determines" the past in exactly the same sense in which the past "determines" the future. The word "determine", here, has a purely logical significance: a certain number of variables "determine" another variable if that other variable is a function of them.
There's also an interesting discussion of this paper in the chapter "What Russell Got Right" in the book Causation, Physics, and the Constitution of Reality: Russell's Republic Revisited, reviewed here (this book is more generally a good reference to check out for anyone interested in the question of whether the idea of 'cause' makes sense in light of modern science).