I have an old memory of my grandmother being scolded because she only wanted to take the paper of the guy she wanted to vote for. And the assessors at my local voting place usually complain when people pick only one paper and want to bypass the voting booth (but it is not like they will come to blows over this).

Is it difficult or utterly impossible to fold the other card and put in a pocket and show it to somebody else outside the polling station?

That would open hacks like "pick my card and my worst opponent one, vote for me and show me my opponent card. And pray that I get at least as many votes as the cards of my opponent that people bring to me."