Let \({\mathcal A}\) denote an \(n\)-dimensional real affine space and \(I\) a real interval. Consider a function \(\Phi:I\to{\mathcal A}\). The osculating flat of order \(l\) of \(\Phi\) at \(x\in I\) (at which \(\Phi\) is \(l\) times differentiable) is \[ \text{Osc}_l \Phi(x):=\bigl\{\Phi (x)+\lambda_1 \Phi'(x)+ \cdots+\lambda_l \Phi^{( l)}\mid \lambda_1,\dots, \lambda_l\in \mathbb{R}\bigr\}. \] \(\Phi\) is defined to be a quasi-Chebyshev function of order \(n\) on \(I\) if it is \(C^{n-1}\) and for all distinct \(\tau_1,\dots,\tau_r\in I\) and all positive integers \(\mu_1,\dots, \mu_r\) such that \(\sum^r_{i=1} \mu_i=n\), the intersection \[ \cap^r_{i=1} \text{Osc}_{n-\mu_i} \Phi(\tau_i) \] is a single point, denoted \(\varphi (\tau_1^{\mu_1}, \dots,\tau_r^{\mu_r})\) (where \(\tau^\mu\) here stands for the \(\mu\)-tuple \((\tau,\dots,\tau))\). If \((x_1,\dots,x_n)\) is a permutation of \((\tau_1^{\mu_1}, \dots,\tau_r^{\mu_r})\) then \[ \varphi(x_1, \dots,x_n):= \varphi(\tau_1^{\mu_1}, \dots,\tau_r^{\mu_r}). \] The symmetric function \(\varphi :I^n\to {\mathcal A}\) so defined is the blossom of \(\Phi\). The paper extends the theory of blossoming (introduced in \textit{L. Ramshaw}, Comput. Aided Geom. Des. 6, No. 4, 323-358 (1989; Zbl 0705.65008) see also earlier papers of the second author) to quasi-Chebyshev functions. Under the further condition that \(\Phi' (x), \dots,\Phi^{(n-1)}(x)\), \(\Phi^{(s)}(x)\) are linearly independent for some \(s\geq n\) for each \(x\in I\), the technical results required for the definition of Bernstein and B-spline bases, and for the development of the de Casteljau and de Boor algorithms are obtained. In Section 3 it is shown that examples of quasi-Chebyshev functions are provided, in a familiar way, by solutions of differential equations. Section 4 is concerned with specific examples in which \(\Phi:[0,1] \to\mathbb{R}^n\) is of the form \[ \Phi(x)= (x,\dots,x^{n-2}, x^{n-1+m_1}, \;(1-x)^{n-1+m_2})^T. \] The results are illustrated (in two senses) by the cases \(n=3,4\) and \(m_1=m_2\).