The angle (theta(Bn)) between the normal to an interplanetary shock front and the upstream magnetic field direction, though often thought of as a property "of the shock," is also determined by the configuration of the magnetic field immediately upstream of the shock. We investigate the interplanetary circumstances of 105 near-Earth quasi-perpendicular shocks during 1996-2005 identified by theta(Bn) >= 80 degrees and/ or by evidence of shock drift particle acceleration. Around 87% of these shocks were driven by interplanetary coronal mass ejections (ICMEs); the remainder were probably the forward shocks of corotating interaction regions. For around half of the shocks, the upstream field was approximately perpendicular to the radial direction, either east-west or west-east or highly inclined to the ecliptic. Such field directions will give quasi-perpendicular configurations for radially propagating shocks. Around 30% of the shocks were propagating through, or closely followed, ICMEs at the time of observation. Another quarter were propagating through the heliospheric plasma sheet (HPS), and a further quarter occurred in slow solar wind that did not have characteristics of the HPS. Around 11% were observed in high-speed streams, and 7% in the sheaths following other shocks. The fraction of shocks found in high-speed streams is around a third of that expected based on the fraction of the time when such streams were observed at Earth. Quasi-perpendicular shocks are found traveling through ICMEs around 2-3 times more frequently than expected. In addition, shocks propagating through ICMEs are more likely to have larger values of theta(Bn) than shocks outside ICMEs.