We propose a joint analysis simultaneously analyzing recurrent and nonrecurrent events subject to general types of interval censoring. The proposed analysis allows for general semiparametric models, including the classes of Box-Cox transformation and inverse Box-Cox transformation models for the recurrent and nonrecurrent events, respectively. A frailty variable is used to account for the potential dependence between the recurrent and non-recurrent event processes. We apply the pseudo likelihood for interval-censored recurrent event data, usually termed as panel count data, and the sufficient likelihood for interval-censored non-recurrent event data. Conditioning on the sufficient statistic for the frailty, and using the working assumption of independence over examination times, the sufficient likelihood does not rely on distributional assumptions on the frailty, and can deal with general interval censorship. We illustrate the proposed methodology by a joint analysis of the numbers of adverse events and time to premature withdrawal from study medication based on a scleroderma lung disease dataset.