These are easy under Linux, but require care more code and care under other operating systems.  They can be implemented portably though!

I have separate Windows versions because they can easily support UTF-8, since timezone names are interestingly kept _only_ in Unicode by Windows.

The Windows version should use GetTimeZoneInformationForYear under Vista or later, but I don't have a machine that has it for testing.

An automated test is difficult, here is what I used:

#include <glib.h>

int main(int argc, char **argv)
{
  GDate *date = g_date_new_dmy (14, 4, 2010);
  GTimeVal tv;

  printf ("%d-%d-%d\n", g_date_get_day (date), g_date_get_month (date), g_date_get_year (date));
  g_date_get_time_val (date, &tv);
  g_date_set_time_val (date, &tv);
  printf ("%d-%d-%d\n", g_date_get_day (date), g_date_get_month (date), g_date_get_year (date));
  printf ("%s %d\n", g_time_val_timezone_name (&tv), g_time_val_timezone_bias (&tv));

  g_date_set_month (date, 2);
  g_date_get_time_val (date, &tv);
  g_date_set_time_val (date, &tv);
  printf ("%d-%d-%d\n", g_date_get_day (date), g_date_get_month (date), g_date_get_year (date));
  printf ("%s %d\n", g_time_val_timezone_name (&tv), g_time_val_timezone_bias (&tv));
}