The extended 22-year solar cycle phenomenon, discovered in observations of the solar corona and variations of the solar differential rotation (torsional oscillations), represents a fundamental heliophysics problem linked to dynamo processes inside the Sun. As observed on the surface, the extended solar cycle starts during a sunspot maximum at high latitudes and consists of a relatively short polar branch (described as “rush to the poles”) and a long equatorward branch that continues through the solar minimum and the next sunspot cycle. Helioseismic observations of the internal dynamics of the Sun during the last two solar activity cycles allow us to identify the dynamical processes associated with the extended solar cycle throughout the depth of the convective zone and to link them with dynamo models. Observational data obtained from the SoHO (1996-2010) and SDO (2010-2020) spacecraft represent measurements of the internal differential rotation, meridional circulation, and thermodynamic parameters. The data indicate that the development of a new extended solar cycle begins at about 60 degrees latitude at the base of the convective zone during the maximum of the previous cycle. Then, the process of magnetic field migration to the Sun’s surface is divided into two branches: fast (in 1-2 years) migration to the poles in the high-latitude zone and slow migration to the equator at middle and low latitudes for ~ 10 years. The subsurface rotational shear layer (leptocline) plays a key role in the formation of the magnetic butterfly diagram. Both the zonal flows (torsional oscillations) and the meridional circulation reveal the 22-year pattern of the extended solar cycle. A self-consistent MHD model of the solar dynamo developed in the mean-field theory framework is in good qualitative and quantitative agreement with the helioseismic observations. The model shows that the extended solar-cycle phenomenon is caused by magnetic field quenching of the convective heat flux and modulation of the meridional circulation induced by the heat flux variations. The model explains why the solar minimum polar field predicts the next sunspot maximum and points to new possibilities for predicting solar cycles from helioseismological data.