Linear sparse arrays with fourth-order cumulant processing can resolve $\mathcal{O}(N^{4})$ directions-of-arrival (DOAs) using $N$ physical sensors, provided that the fourth-order difference co-array $\Delta_{4}$ contains a contiguous segment of size $\mathcal{O}(N^{4})$. Furthermore, if $\Delta_{4}$ has no holes, then the received data can be fully exploited in subspace-based DOA estimators. However, few existing arrays attain large hole-free $\Delta_{4}$. Many existing arrays designed for $\Delta_{4}$ are constructed from two smaller arrays, called the basis arrays. Nevertheless, such arrays either restrict the basis arrays to certain types or have no guarantee of hole-free $\Delta_{4}$. This paper proposes the half-inverted (HI) arrays, parameterized by two basis arrays $\mathbb{S}^{(1)}$ and $\mathbb{S}^{(2)}$, the shifting parameter $M$, and the scaling parameter $\sigma$. An HI array consists of $\mathbb{S}^{(1)}$ and an inverted, scaled, and shifted version of $\mathbb{S}^{(2)}$. HI arrays are guaranteed with hole-free $\Delta_{4}$ over a range of $(M,\sigma)$ pairs. This property unifies several existing arrays with hole-free $\Delta_{4}$ and admits an optimization problem over $(M,\sigma)$. The half-inverted general hole-free (HIGH) scheme is defined as the HI array with a closed-form and optimized $(M,\sigma)$ pair determined by the second-order co-arrays of the basis arrays. The HIGH scheme enjoys a large hole-free $\Delta_{4}$. The shift-scale representation (SSR) is presented to study $\Delta_{4}$ of HI arrays visually. From these results, the half-inverted array based on second-order optimization and extended shift (HI-SOES) is proposed. For a fixed $N$, HI-SOES synthesizes a hole-free $\Delta_{4}$ larger than an existing array. Numerical examples demonstrate the DOA estimation performance of HI arrays and existing arrays.