Many socially consequential beliefs, notably political and religious ideologies, consist not of single propositions in isolation from others but as systems of many propositions. Philip Converse, one of the most influential political scientists of the twentieth century, proposed that such systems can be understood as networks of propositions and predicted that they would be highly intercorrelated in those with strong ideological commitments but less so in people who are less ideological. We used recent advances in network psychometrics to test this account in relation to the political beliefs of a representative sample of 2,058 UK adults, who rated themselves on the left-right dimension and then reported their attitudes toward 18 policy issues. We divided participants into equally-sized groups of left-wing, centrist and right-wing participants and found that, as Converse had predicted, the networks of those at either end of the left-right continuum were similar in structure, being significantly more interconnected than the networks of those who identified themselves as centrists, even though the actual beliefs were (for the most part) polar opposites. This finding, which was robust to sensitivity checks, aligns with previous research which has shown that people at the political extremes, compared to those in the centre, are more certain about their beliefs and less likely to change them over time. In each ideological group we also identified the same three communities of beliefs which mapped onto classic accounts of authoritarian attitudes, altruism and cooperativeness, and personal liberty. Attitudes towards gay rights had the highest predictability index in all three networks and was the most central node in the right and centre networks, suggesting that these attitudes play a largely unrecognised but important role in ideological positioning. Our analytical approach has implications for not only political beliefs but all organized belief systems.