An endogenous magnetic reconnection process is characterized by a driving factor that lays within the layer where a drastic change of magnetic field topology occurs. This kind of process is shown to take place in the presence of an electron temperature gradient in a well-confined plasma where, referring to quasi-collisionless regimes, the resulting electron temperature fluctuations can be anisotropic. Then a class of (radially) localized reconnecting modes is identified. These involve a transverse generated field of odd parity (as a function of the radial variable), and have finite (phase) velocities of propagation contrary to commonly considered reconnecting modes. The widths of the relevant reconnection layers remain significant even when large macroscopic distances are considered. Given that there are plasmas in the Universe with considerable electron thermal energy contents, these features can be relied upon in order to produce magnetic field generation, or conversion of magnetic energy into particle energy when the coupling of the localized odd modes to extended even modes can be significant. In any case, the resulting magnetic islands are not symmetric. With their excitation these modes can extract momentum from the main body of the plasma column which should recoil in the opposite direction. The excitation of antisymmetric endogenous modes is shown to be relevant to the electron temperature heating due to the reaction products in a fusion burning plasma as, in this case, the longitudinal thermal conductivity on selected rational magnetic surfaces can be decreased, relative to its collisional value, by the effects of reconnection. This kind of steepening is proposed to have a role in enhancing the growth rate of the instability involved in disruption events of the plasma column. The best agreement between theory and experiments concerning the onset of magnetic reconnection is (probably) represented by the theory of the internal kink mode. The observed accelerated reconnection rate following the onset is suggested as being explained by the formation of a relatively large magnetic island with a local steepening of the electron temperature gradient. A new kind of odd 'thermonuclear heating' mode associated with symmetric reconnection is identified.