We introduce two-player turn-based zero-sum register games on an infinite linearly ordered data domain. Register games have a finite set of registers intended to store data values. At each round, Adam picks some data in the domain, which is tested against the data contained in the registers, using the linear order. Depending on which test holds (exactly one is required), Eve decides to assign, or not, the data to some of her registers, and the game deterministically evolves to a successor vertex depending on her assignment choice. Eve wins the game if she has a strategy which depends only on the tests that hold (and not on the concrete data values of Adam), such that whichever values Adam provides, the sequence of visited vertices of the game satisfies some parity condition. We show the decidability of register games over data domains N and Q. For Q, they can be solved in ExpTime and finite-memory strategies always suffice to win. For N, we show that deciding the existence of a finite-memory strategy is also in ExpTime. We apply these results to solve the synthesis problem of strategies resolving non-determinism in (non-deterministic) register transducers on data words.