Voting for a representative and voting directly on an issue are two very different things. Sure, the idea behind them is the same, but the level of transparency

Voting for a representative and voting directly on an issue are two very different things. Sure, the idea behind them is the same, but the level of transparency of your stance is drastically higher in the latter case. In other words, you can always say "Well, yeah, I know John Doe is 'pro-choice', but I didn't vote for him because of that. I voted for him because of his ideas for lowering the unemployment." On the other hand, if you publicly voted "pro-choice" instead of "pro-life", there's no way you can deny it. So it comes down to plausible deniability. As to why you would want to deny your own choice in the first place, several other commenters have already explained that.