Holmstedt's formula for the $K$-functional: the limit case ${\theta }_0={\theta }_1$

We consider $K$-interpolation spaces involving slowly varying functions, and derive necessary and sufficient conditions for a Holmstedt-type formula to be held in the limiting case ${\theta }_0={\theta }_1\in \{0,1\}\mathrm{.}$ We also study the case ${\theta }_0={\theta }_1\in (0,1)\mathrm{.}$ Applications are given to Lorentz-Karamata spaces, generalized gamma spaces and Besov spaces.