The tiles for the NZ map service has been created in a different projection. You can see this at http://services.arcgisonline.co.nz/ArcGIS/rest/services/LINZ/geotiffs/MapServer where it says "Spatial Reference: 2193".The flex viewer does not re-project cached services.You have two options, the best one is to just use services in that projection, for example:<map wraparound180="true" top="40">
<basemaps>
<layer label="NZ Topo" type="tiled" visible="true" url="http://services.arcgisonline.co.nz/ArcGIS/rest/services/LINZ/geotiffs/MapServer"/>
</basemaps>
The other solution is to request a dynamicly created map from the LINZ service. This will not look as good, nor perform as well:<map wraparound180="true" top="40">
<basemaps>
<layer label="Streets" type="tiled" visible="true"
url="http://server.arcgisonline.com/ArcGIS/rest/services/World_Street_Map/MapServer"/>
<layer label="NZ Topo" type="dynamic" visible="false"
url="http://services.arcgisonline.co.nz/ArcGIS/rest/services/LINZ/geotiffs/MapServer"/>
<layer label="Aerial" type="tiled" visible="false"
url="http://server.arcgisonline.com/ArcGIS/rest/services/World_Imagery/MapServer"/>
<layer label="Topo" type="tiled" visible="false"
url="http://server.arcgisonline.com/ArcGIS/rest/services/World_Topo_Map/MapServer"/>
</basemaps>
(make sure that the NZ service is not listed first, in which case the map will use the LINZ projection and the tiled from ArcGIS Online services will not work)