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I have mean response times in ms to either congruent or incongruent implied hand action direction words and required direction of response. There are four sets of results, two congruous and two incongruous, with a small difference between the means in each paired set. Is there any reason that I would run a repeat measures Anova to analyse these results or is a paired-samples t-test, run twice for each set of data sufficient, particularly if I then use Cohen's d and confidence intervals?

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The ANOVA would let you test the interaction. That is, it would provide a test of whether the difference between congruous and incongruous differs as a function of direction.

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