Properties of Heaviside Function

Question

Let $H(x)$ be the Heaviside function defined by

\begin{cases} 1 & \text{if } x\geq0\\ 0 & \text{if } x<0 \end{cases}

I know that

1. $H'(x)=\delta(x)$. The derivative of the Heaviside function is the delta function.
2. $\delta(x)=\delta(-x)$. The delta function is symetric.

Does

1. $H(x)=H(-x)$?
2. $H(x)=-H(x)$?

It appears that

$$ -\delta(x)\delta(-y)=\delta(x)\delta(y)$$

and

$$ -\delta(-x)\delta(y)=\delta(x)\delta(y)$$

Do both of these properties follow from the definition of the Heaviside function?

Answer 1

No, $H(x)=1-H(-x)$ for $x\ne 0$. Integrating something even gives something odd plus an integration constant.