The authors consider the following neutral differential difference equation \[ [x(t)-P(t)x(t-\tau)]^{(n)}+Q(t)x(t-\sigma)=0,\quad t\geq t_0, \] where \(n\) is an odd number, \(P\in C([t_0,\infty),{\mathbb{R}})\), \(Q\in C(t_0,\infty),{\mathbb{R}}^+)\), \(\tau=\text{ const}>0\) and \(\sigma=\text{ const} \geq 0\). New criteria for the oscillation of all solutions are given when \(P(t)-1\) is allowed to be oscillatory. Sufficient conditions for the equation to have a positive solution when \(P(t)-1\) is allowed to oscillate are presented. The main assumption for the oscillatory behavior looks as follows. Let us denote \(\bigcup_{i\in\{0,1,2,\ldots\}}\{s: t_1+i\tau\leq s\leq t_2+i\tau\}\) by \(E[t_1,t_2]\). It is assumed that there exist \(t^*\geq t_0\), \(t_2>t_1\geq t_0\), \(\alpha\geq 1\) and a nondecreasing function \(H\in C([\min\{t_1,\tau\},\infty),(0,\infty))\) such that \[ P(t^*+i\tau)\leq 1 \text{ for all integers }n, \tag{1} \]
\[ P(t)\geq\alpha\text{ for all }t\in E[t_1,t_2], \]
\[ H(s)+H(t)\geq H(s+t)\text{ for } s,t\geq\min\{t_1,\tau\}, \] and \[ \int_{E[t_1+\sigma,t_2+\sigma]}Q(s)\exp[H(s)\ln\alpha/H(\tau)] ds=\infty. \] The assumption (1) can be replaced by the following: \(P(t)\) is oscillatory, or there exists \(s_0\geq t_0\) such that \(P(t)>0\) for \(t\geq s_0\) and \[ \sum_{k=1}^{\infty}[P(s_0+\tau)P(s_0+2\tau)\cdots P(s_0+k\tau)]^{-1}=\infty. \]