Cancer cell invasion is recognised as one of the hallmarks of cancer and involves several inner-related multiscale processes that ultimately contribute to its spread into the surrounding tissue. In order to gain a deeper understanding of the tumour invasion process, we pay special attention to the interacting dynamics between the cancer cell population and various constituents of the surrounding tumour microenvironment. To that end, we consider the key role that ECM plays within the human body tissue, providing not only structure and support to surrounding cells, but also acting as a platform for cells communication and spatial movement. There are several other vital structures within the ECM, however we are going to focus primarily on fibrous proteins, such as fibronectin. These fibres play a crucial role in tumour progression, enabling the anchorage of tumour cells to the ECM. In this work we consider the two-scale dynamic cross-talk between cancer cells and a two component ECM (consisting of both a fibre and a non-fibre phase). To that end, we incorporate the interlinked two-scale dynamics of cells-ECM interactions within the tumour support that contributes simultaneously both to cell-adhesion and to the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is embedded within a multiscale moving boundary approach for the invading cancer cell population, in the presence of cell-adhesion at the tissue scale and cell-scale fibre redistribution activity and leading edge matrix degrading enzyme molecular proteolytic processes. The overall modelling framework will be accompanied by computational results that will explore the impact on cancer invasion patterns of different levels of cell adhesion in conjunction with the continuous ECM fibres rearrangement.