In this paper, our goal is to investigate to what degree multilingual pretrained language models capture cross-linguistically valid abstract linguistic representations. We take the approach of developing curated synthetic data on a large scale, with specific properties, and using them to study sentence representations built using pretrained language models. We use a new multiple-choice task and datasets, Blackbird Language Matrices (BLMs), to focus on a specific grammatical structural phenomenon -- subject-verb agreement across a variety of sentence structures -- in several languages. Finding a solution to this task requires a system detecting complex linguistic patterns and paradigms in text representations. Using a two-level architecture that solves the problem in two steps -- detect syntactic objects and their properties in individual sentences, and find patterns across an input sequence of sentences -- we show that despite having been trained on multilingual texts in a consistent manner, multilingual pretrained language models have language-specific differences, and syntactic structure is not shared, even across closely related languages.