When an organization is the author and the publisher of a work, the handbook advises writers to begin the works-cited-list entry with the title of the source. Is it OK to use the organization’s name in a signal phrase or in the in-text citation even though the name does not begin the works-cited-list entry?

No. The text should always key to the list of works cited. You can provide the key in the parenthetical citation or in your text. Below are two acceptable ways to cite the MLA Handbook:

According to the Modern Language Association of America, documentation should be useful to readers (MLA Handbook 4).

According to the MLA Handbook, documentation should be useful to readers (4).

The following two examples are insufficient, because MLA Handbook, the title of the work that begins the works-cited-list entry, is not mentioned:

Documentation should be useful to readers (Modern Language Association 4).

The Modern Language Association of America says that documentation should be useful to readers (4).

Work Cited
MLA Handbook. 8th ed., Modern Language Association of America, 2016.

Citing sources in your text is covered in detail in chapter 6 of the handbook.