Summary. With the ready availability of spatial databases and geographical information system software, statisticians are increasingly encountering multivariate modelling settings featuring associations of more than one type: spatial associations between data locations and associations between the variables within the locations. Although flexible modelling of multivariate point-referenced data has recently been addressed by using a linear model of co-regionalization, existing methods for multivariate areal data typically suffer from unnecessary restrictions on the covariance structure or undesirable dependence on the conditioning order of the variables. We propose a class of Bayesian hierarchical models for multivariate areal data that avoids these restrictions, permitting flexible and order-free modelling of correlations both between variables and across areal units. Our framework encompasses a rich class of multivariate conditionally autoregressive models that are computationally feasible via modern Markov chain Monte Carlo methods. We illustrate the strengths of our approach over existing models by using simulation studies and also offer a real data application involving annual lung, larynx and oesophageal cancer death-rates in Minnesota counties between 1990 and 2000.