We present high time resolution polarimetric profiles of 27 predominantly southern hemisphere millisecond pulsars, 15 of which have no previously published polarimetry. These observations were made with a new 128-MHz baseband recorder at the Parkes Observatory. There has been some suggestion that millisecond pulsar profiles can undergo radical changes in both pulse shape and polarimetry, mainly due to discrepancies between the Bonn and Jodrell Bank polarimetric studies. If millisecond pulsars are intrinsically unstable, this has ramifications for precision timing and the millisecond pulsar emission mechanism. However, we find ourselves in good agreement with the Jodrell Bank data, and, in most cases, very poor agreement with the Bonn results. The presented polarimetric observations do display some phenomena common to those displayed by normal pulsars, including orthogonal mode transitions in position angle and associated sense changes of circular polarization. The behaviour of the position angle presented by these pulsars is, with some significant exceptions, a shallow or flat sweep across the pulse. This behaviour lends support to theories that suggest millisecond pulsar emission regions are wider, at least in terms of pulse longitude, than those of the normal pulsars. The broad millisecond pulsars J2124-3358 and J2145-0750 display position-angle behaviour that departs significantly from that expected if the magnetic field of these pulsars has a simple dipolar structure.