One of the fundamental structural properties of many networks is triangle closure. Whereas the influence of this transitivity on a variety of contagion dynamics has been previously explored, existing models of coevolving or adaptive network systems use rewiring rules that randomize away this important property. In contrast, we study here a modified coevolving voter model dynamics that explicitly reinforces and maintains such clustering. Employing extensive numerical simulations, we establish that the transitions and dynamical states observed in coevolving voter model networks without clustering are altered by reinforcing transitivity in the model. We then use a semi-analytical framework in terms of approximate master equations to predict the dynamical behaviors of the model for a variety of parameter settings. PACS numbers: 89.75.Hc, 87.23.Ge, 64.60.aq, 89.75.Fb The study of dynamics on networks has led to a number of successes identifying how the structure of the underlying network impacts the dynamics occurring on the network [1][2][3] and whether dynamics taking place on the network also promotes organizing features of the network structure itself [4][5][6]. Within this larger research theme, significant attention has been given to exploring the role of network structures on the spread of contagions and opinions [1,7,8,10], including efforts to understand and quantify features in the spread of contagions due to different local and global structural properties [11,12]. The study of opinions spreading in social networks has gained additional interest due to the rise of social media and its role in mobilizing and framing public opinion [13], including elections and advertising campaigns [14]. Hence, understanding and quantifying the interplay between network structures and contagion dynamics is of very broad interest and scope [1].The processes involved in collective opinion formation, including the role of network properties in these processes, are extremely complex [15]. We thus aim to study the properties emerging from a simple local model for interaction that incorporates only some of the essential features involved. The coevolving voter model is one of the simplest and most studied generic models for the interplay between opinion formation and the network. In this model, connected nodes with discordant opinions are resolved by one neighbor in the pair either changing its opinion or dropping the connection (in favor of a newly rewired connection to another node in the network). This model reproduces several complex features observed in collective opinion formation and has led to a variety of computational and analytical results on different aspects of the model [1,2,10,15,16,18]. However, all previous variants of this model (including those studied by the present authors) have ignored one of the most fundamental features of networks, namely the higher propensity for a connection between two nodes that are both already connected to a third node, closing the triangle between them [5]. Specifically, the rewi...