ArcPad would just be reporting the HAE, in meters, as reported by your receiver -- UNLESS you have values set in the GPS Height tab of GPS Preferences. Values you set there will affect the reported elevation.
Now - is your map datum WGS 1984 or is it NAD 1983? If it's WGS 1984, then ArcPad is absolutely not adjusting the elevation from the HAE reported directly from the receiver (again, unless you have values set in the GPS Height tab).
If it's NAD 1983, there there will be a very slight elevation adjustment with the datum transform, but on the order of mm (maybe cm) since GRS-80 and WGS 1984 ellipsoids are nearly identical.
I would love to help you dig in deeper if you want me to.
Also, please note that we wrote an extension to ArcPad that handles the high accuracy WGS 1984 to NAD 1983 (2011) transformation as well as HAE to MSL elevations using Geoid12a.
The extension is called GeoBullseye: Mobile GIS Experts : GeoMobile Innovations : GeoBullseye For ArcPad