Consider \(L_2(\mathbb{D})\), where \(\mathbb{D}\) is the unit disk on the complex plane, and its Bergman subspace \(L^2_a\) consisting of analytic functions. Let \(P\) be the orthogonal projection of \(L_2(\mathbb{D})\) onto \(L^2_a\). Given \(\varphi \in L^{\infty}(\mathbb{D})\), the dual Toeplitz operator \(S_{\varphi}\) with symbol \(\varphi\) is defined on \((L^2_a)^{\perp}\) as \(S_{\varphi}u = (I-P)(\varphi u)\). The aim of the paper is to study the properties of dual Toeplitz operators with continuous and bounded harmonic symbols. The authors construct a class of continuous functions such that dual Toeplitz operators with these functions as the symbols have disconnected spectra, characterize the hyponormality of dual Toeplitz operators with bounded harmonic symbols, and show that the spectra of dual Toeplitz operators with certain bounded harmonic symbols are connected (cf. [\textit{K. Stroethoff} and \textit{D. Zheng}, Trans. Am. Math. Soc. 354, No. 6, 2495--2520 (2002; Zbl 0996.47036)]). In addition, they establish several sufficient conditions for dual Toeplitz operators to be invertible.