The men's side and women's side of the sauna switches back and forth depending on whether it's an odd or even day of the month.
この場合、次のような言い方ができますよ。
ーThe men's side and women's side of the sauna switches back and forth depending on whether it's an odd or even day of the month.
「サウナの男性側と女性側は奇数日と偶数日かによって入れ替わる」
to switch back and forth で「入れ替わる・切り替わる」などと言えます。
an odd day (of the month) で「奇数日」
an even day (of the month) で「偶数日」
ご参考まで！
カナダ 
