  Period-luminosity (PL) sequences of long period variables (LPVs) are commonly
interpreted as different pulsation modes, but there is disagreement on the
modal assignment. Here, we re-examine the observed PL sequences in the Large
Magellanic Cloud, including the sequence of long secondary periods (LSPs), and
their associated pulsation modes. Firstly, we theoretically model the sequences
using linear, radial, non-adiabatic pulsation models and a population synthesis
model of the LMC red giants. Then, we use a semi-empirical approach to assign
modes to the pulsation sequences by exploiting observed multi-mode pulsators.
As a result of the combined approaches, we consistently find that sequences B
and C$^{\prime}$ both correspond to first overtone pulsation, although there
are some fundamental mode pulsators at low luminosities on both sequences. The
masses of these fundamental mode pulsators are larger at a given luminosity
than the mass of the first overtone pulsators. These two sequences B and
C$^{\prime}$ are separated by a small period interval in which large amplitude
pulsation in a long secondary period (sequence D variability) occurs, meaning
that the first overtone pulsation is not seen as the primary mode of pulsation.
Observationally, this leads to the splitting of the first overtone pulsation
sequence into the two observed sequences B and C$^{\prime}$. Our two
independent examinations also show that sequences A$^{\prime}$, A and C
correspond to third overtone, second overtone and fundamental mode pulsation,
respectively.
