Standard Mendelian randomisation analysis can produce biased results if the genetic variant defining the instrumental variable (IV) is confounded and/or has a horizontal pleiotropic effect on the outcome of interest not mediated by the treatment. We provide novel identification conditions for the causal effect of a treatment in the presence of unmeasured confounding, by leveraging an invalid IV for which both the IV independence and exclusion restriction assumptions may be violated. The proposed Mendelian Randomization Mixed-Scale Treatment Effect Robust Identification (MR MiSTERI) approach relies on (i) an assumption that the treatment effect does not vary with the invalid IV on the additive scale; and (ii) that the selection bias due to confounding does not vary with the invalid IV on the odds ratio scale; and (iii) that the residual variance for the outcome is heteroscedastic and thus varies with the invalid IV. Although assumptions (i) and (ii) have, respectively appeared in the IV literature, assumption (iii) has not; we formally establish that their conjunction can identify a causal effect even with an invalid IV subject to pleiotropy. MR MiSTERI is shown to be particularly advantageous in the presence of pervasive heterogeneity of pleiotropic effects on additive scale, a setting in which two recently proposed robust estimation methods MR GxE and MR GENIUS can be severely biased. For estimation, we propose a simple and consistent three-stage estimator that can be used as preliminary estimator to a carefully constructed one-step-update estimator, which is guaranteed to be more efficient under the assumed model. In order to incorporate multiple, possibly correlated and weak IVs, a common challenge in MR studies, we develop a MAny Weak Invalid Instruments (MR MaWII MiSTERI) approach for strengthened identification and improved accuracy. We have developed an R package MR-MiSTERI for public use of all proposed methods. We illustrate MR MiSTERI in an application using UK Biobank data to evaluate the causal relationship between body mass index and glucose, thus obtaining inferences that are robust to unmeasured confounding, leveraging many weak and potentially invalid candidate genetic IVs. MaWII MiSTERI is shown to be robust to horizontal pleiotropy, violation of IV independence assumption and weak IV bias. Both simulation studies and real data analysis results demonstrate the robustness of the proposed MR MiSTERI methods.