Lucy if you have a copy of the GPX Validator widget that Danial is using for DIY validations - that shows total climb and is a nice little one-click pony. (But it requires .NET to be installed which makes it no use to me.)
Otherwise GPSAltitudeChart mentioned above does the job - free download but requires Java to be installed.
But the problem is that the numbers you get must always be regarded as quite iffy. If you do it for your daily work trip it would be interesting (well I think so!) to see how much variation you get.
Ignoring for the moment that you have a barometric GPS -
There is usually a big 'spike' associated with switch-on at the start of the day and if (like most people would do) you turn it on and set off on your ride almost immediately - it can take several minutes for the GPS altitude information to stabilise, even after your position on the map seems to be OK. Subsequently various factors can cause further spikes (both up and down) which pass unnoticed while riding but may sometimes get written into the tracklog. All these things would introduce errors into the total climb figure, I suppose tending to increase it.
The barometric GPS is supposed to improve the situation but I think its highly arguable whether it actually does. It certainly smooths the spikes and it is the barometric altitude figure, not the GPS one, that gets written into the tracklog.
But the barometer has to be calibrated, so (with default settings) it does this automatically (using the GPS current best guess) at some point shortly after startup. This is not good - because (as above) the altitude information takes a long time after startup to stabilise. Then the barometer continues to auto-calibrate (every 15 minutes, according to Garmin) using the current GPS-derived figure. Given that the whole point of the barometer is that it is deemed to be more accurate than the GPS, it seems a little odd to me that the (accurate) barometer is synched with the (inaccurate) GPS on such a frequent basis - if the GPS is truly that unreliable, each synch is going to introduce a step in the altitude plot. These steps will again affect total climb calculations.
Personally I don't think the GPS-derived data is as bad as all that, and in fact I've turned the barometer off (that is, set it to 'stationary' mode, so it just acts like a weather barometer - which is itself quite useful on holiday) and prefer to just have the GPS data. In this mode (but only this mode) the heights written into the tracklog are the GPS-derived ones.