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Given    $$E_0, E_1, F_0, F_1, E$$  E 0 , E 1 , F 0 , F 1 , E rearrangement invariant function spaces,    $$a_0$$  a 0 ,    $$a_1$$  a 1 ,    $$\mathrm {b}_0$$  b 0 ,    $$\mathrm {b}_1$$  b 1 ,    $$\mathrm {b}$$  b slowly varying functions and    $$0&lt; \theta _0&lt;\theta _1&lt;1$$  0 &lt; θ 0 &lt; θ 1 &lt; 1 , we characterize the interpolation spaces    $$\begin{aligned} \big ({\overline{X}}^{{\mathcal {R}}}_{\theta _0,\mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{{\mathcal {R}}}_{\theta _1, \mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E},\quad \big ({\overline{X}}^{{\mathcal {L}}}_{\theta _0, \mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{\mathcal L}_{\theta _1,\mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E} \end{aligned}$$  ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 R , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 R ) θ , b , E , ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 L , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 L ) θ , b , E and    $$\begin{aligned} \big ({\overline{X}}^{{\mathcal {R}}}_{\theta _0,\mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{{\mathcal {L}}}_{\theta _1, \mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E},\quad \big ({\overline{X}}^{{\mathcal {L}}}_{\theta _0, \mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{\mathcal R}_{\theta _1,\mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E}, \end{aligned}$$  ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 R , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 L ) θ , b , E , ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 L , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 R ) θ , b , E , for all possible values of    $$\theta \in [0,1]$$  θ ∈ [ 0 , 1 ] . Applications to interpolation identities for grand and small Lebesgue spaces, Gamma spaces and  A  and  B -type spaces are given.

positivity

General reiteration theorems for $${{\mathcal {R}}}$$ and $${{\mathcal {L}}}$$ classes: mixed interpolation of $${{\mathcal {R}}}$$ and $${{\mathcal {L}}}$$-spaces