Distributed infrastructures for computation and analytics are now evolving towards an interconnected ecosystem allowing complex scientific workflows to be executed across hybrid systems spanning from IoT Edge devices to Clouds, and sometimes to supercomputers (the Computing Continuum). Understanding the performance trade-offs of large-scale workflows deployed on such complex Edge-to-Cloud Continuum is challenging. To achieve this, one needs to systematically perform experiments, to enable their reproducibility and allow other researchers to replicate the study and the obtained conclusions on different infrastructures. This breaks down to the tedious process of reconciling the numerous experimental requirements and constraints with low-level infrastructure design choices. To address the limitations of the main state-of-the-art approaches for distributed, collaborative experimentation, such as Google Colab, Kaggle, and Code Ocean, we propose KheOps, a collaborative environment specifically designed to enable cost-effective reproducibility and replicability of Edge-to-Cloud experiments. KheOps is composed of three core elements: (1) an experiment repository; (2) a notebook environment; and (3) a multi-platform experiment methodology. We illustrate KheOps with a real-life Edge-to-Cloud application. The evaluations explore the point of view of the authors of an experiment described in an article (who aim to make their experiments reproducible) and the perspective of their readers (who aim to replicate the experiment). The results show how KheOps helps authors to systematically perform repeatable and reproducible experiments on the Grid5000 + FIT IoT LAB testbeds. Furthermore, KheOps helps readers to cost-effectively replicate authors experiments in different infrastructures such as Chameleon Cloud + CHI@Edge testbeds, and obtain the same conclusions with high accuracies (> 88% for all performance metrics).