Circular (directional) data arise whenever observations are measured as angles on the unit circle, such as wind direction, time of day, or calendar phase, and require statistical methods that respect the periodicity of the domain $[0; 2π)$. While circular-linear and linear-circular regression models are well established, regression models for a binary or binomial response observed jointly with a circular predictor remain largely undeveloped, with the sole closely related study restricted to the symmetric logit link. This paper develops and evaluates a circular logistic regression framework in which the linear predictor is expressed through the cosine and sine of the circular covariate, and compares the performance of symmetric link functions (logit, probit) against asymmetric alternatives (complementary log-log, Cauchit, and a skew-logit power link) under a generalized linear model formulation. A Monte Carlo simulation generates circular predictors from the von Mises distribution under two concentration regimes and evaluates model fit using the Akaike Information Criterion (AIC) and deviance. The methodology is illustrated with two real data sets: daily rainfall occurrence and wind direction recorded in Macomb, Illinois, and monthly earthquake counts in Western Anatolia, Turkiye, the latter used to connect the binary circular model to the related circular Poisson regression framework for count outcomes. Results indicate that the choice of link function matters most when the circular predictor is broadly dispersed and the response is markedly unbalanced; under high concentration of the predictor, symmetric links are preferred and asymmetric links are prone to instability. Practical guidelines and directions for future software development are discussed.

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