Disengagement and disenchantment with the Parliamentary process is an important concern in today's Western democracies. Members of Parliament (MPs) in the UK are therefore seeking new ways to engage with citizens, including being on digital platforms such as Twitter. In recent years, nearly all (579 out of 650) MPs have created Twitter accounts, and have amassed huge followings comparable to a sizable fraction of the country's population. This paper seeks to shed light on this phenomenon by examining the volume and nature of the interaction between MPs and citizens. We find that although there is an information overload on MPs, attention on individual MPs is focused during small time windows when something topical may be happening relating to them. MPs manage their interaction strategically, replying selectively to UK-based citizens and thereby serving in their role as elected representatives, and using retweets to spread their party's message. Most promisingly, we find that Twitter opens up new avenues with substantial volumes of cross-party interaction, between MPs of one party and citizens who support (follow) MPs of other parties.