This paper analyses the Whole Earth Telescope observations of HL Tau 76, the first discovered pulsating DA white dwarf. The star was observed during two Whole Earth Telescope campaigns. It was a second priority target during the XCOV13 campaign in 1996 and the first priority one during the XCOV18 campaign in 1999. The 1999 campaign reached 66.5% duty cycle. With a total duration of 18 days, the frequency resolution achieved is 0.68 μHz. With such a frequency resolution, we were able to find as many as 78 significant frequencies in the power spectrum, of which 34 are independent frequencies after removal of all linear combinations. In taking into account other frequencies present during the 1996 WET campaign and those present in earlier data, which do not show up in the 1999 data set, we find a total of 43 independent frequencies. This makes HL Tau 76 the richest ZZ Ceti star in terms of number of observed pulsation modes. We use those pulsation frequencies to determine as much as possible of the internal structure of HL Tau 76. The pulsations in HL Tau 76 cover a wide range of periods between 380 s and 1390 s. We propose an identification for 39 of those 43 frequencies in terms of ℓ=1 and ℓ=2 non-radial g-modes split by rotation. We derive an average rotation period of 2.2 days. The period distribution of HL Tau 76 is best reproduced if the star has a moderately "thick" hydrogen mass fraction log qH ≥ -7.0. The results presented in this paper constitute a starting point for a detailed comparison of the observed periods with the periods calculated for models as representative as possible of HL Tau 76.