The federal government and the Firestone Tire Rubber Co. have signed the final agreement for the recall of several million Firestone "500" and other steel belted radial tires found to be unsafe by the Department of Transportation.

Although recall efforts began Oct. 20, when the two parties reached a verbal agreement, several disagreements over details of the massive recall caused virtual chaos at Firestone dealerships around the country, leaving thousands of consumers confused over whether their tires were covered by the recall.

The signing of the final agreement late Wednesday cleared up the differences between Firestone and the National Highway Traffic Safety Administration, and should, according to NHTSA chief Joan Claybrook, allow the recall to proceed "as expeditiously as possible."

Under the terms of the signed agreement, Firestone will recall and replace, free of charge:

All 5-rib "500" Steel Belted Radials -- including private brands of the same internal construction -- manufactured in the U.S. or Canada and sold on or after Sept. 1, 1975, provided they were manufactured before Jan. 1, 1977.

Five-rib tires have four deep grooves in the tread design. Tires in this first group have 10-digit DOT identification numbers ending with numbers from 355 through 525 or 016 through 526.

All 7-rib "500" Steel Belted Radials -- also including private brands -- and Firestone TPC tires made in the U.S. or Canada and sold on or after Sept. 1, 1975, and manufactured before May 1, 1976.

Seven-rib tires have six deep grooves. Tires in this category have DOT identification numbers ending in 355 through 525 and 016 through 176.

All 5-rib and 7-rib Steel Belted radial "500" and "TPC" tires manufactured before Sept. 1, 1975, which owners can prove were purchased after that date. All Firestone "500" and private brand tires of the same construction shall be considered "sold" after Sept. 1, 1975, and thus eligible for free replacement, if the tires have DOT ID numbers showing they were manufactured on or after March 1, 1975, and were not mounted on 1975 model cars. Tires on 1975 cars will be counted if it can be shown the tires were made after Sept. 1, 1975, or if there is proof of purchase of the tires or vehicle after that date.

The final three digits of the identification numbers are important because they give the week of manufacture. The numbers 156 mean the tire was produced in the 15th week of 1976.

NHTSA said that while possession of tires is sufficient evidence of ownership in many cases of the recall, customers having tires sold after Sept. 1, 1975, but made before March 1, 1975, will have to show other proof of purchase.

Proof of purchase in the case of replacement tires, according to the agreement, can consist of the purchaser's or owner's recall notification letter from Firestone, the new tire invoice, or other evidence of purchase along with the tires being recalled.

In the case of tires purchased with a new car, known as the "TPC" models, proof of purchase, if needed, will consist of a notification letter from Firestone, new vehicle invoice or title, or other evidence of the vehicle's purchase date, and, of course, the recalled tire itself.

One area of disagreement between Firestone and NHTSA had been wornout tires. Under the final settlement, Firestone said it will accept tires that are considered bald -- when they have less than 2/32 of an inch of tread left -- only if the tires are still mounted on a car, and if the individual presenting them for replacement can prove he is the owner of the car or an agent of the owner.

All recalled tires will be replaced by the Firestone 721 steel-belted radial tire, the company's top of the line. But if the consumer wishes, he can choose from among comparable tires which may be available. If the consumer chooses a lower-priced tire, he will be required to sign a document indicating he is satisfied that the company has met its obligation.

Firestone said it will be responsible for the recall of private brand tires produced by the company for other retailers.

Private brands covered by the recall include: Shell Steel Radial, K-Mart 40 steel radials, National Steel Radial, Union Steel Radial, Seiberling RT 78 Steel Radial, Zenith Supreme Steel Radial, Holiday Supreme Steel Radial, Caravelle Supreme and Caravelle Double Steel Radial, JTW Ferrari Supreme Steel Radial, Lemans Steel Belted Radial, JTW Ferrari Steel Radial, Atlas Goldenaire II (only those with a serial number beginning with VD), Wards Grappler II Steel Radial (only those with serial numbers beginning with VD or VK), Wards Grappler Steel Radial (with serial numbers beginning VD, VJ, or VK.)

Asked to recall tires manufactured earlier than the cutoff dates listed above, Firestone refused. The company did, however, agree to offer an exchange of new tires at half price for the earlier "500's", "TPC's" and private brand tires still on the road.

In addition, Firestone has agreed to reimburse, upon presentation of necessary documentation, persons who already paid a pro-rata adjustment price to replace tires that would have been eligible for free replacement under the recall. Forms for such reimbursements will be available shortly at Firestone dealers.

All tires replaced for free or for half price will be mounted and balanced at no charge to the motorist.

Firestone has estimated that it will reach 70 to 80 percent of the owners of the tires affected by the recall with notification letters.

For those who do not receive letters, however, the company has agreed to take out three-quarter page advertisements in 242 major Sunday newpapers in the top 200 advertising markets in the country -- probably on Dec. 17.

In addition, the company will send explanatory press releases to 3,800 other daily and weekly newspapers across the country. And it will prepare and show two 30-second TV commericals with company chairman Richard Riley or other appropriate company spokesman, announcing the recall and referring customers to the newspaper advertisements for details.

The company agreed to show the commericals at least once on each of two national networks at times normally used by the company in its national tire advertising.

NHTSA has reserved the right to review and approve the advertisements.

Firestone has estimated the recall will cover 7.5 million tires for free replacement and another 6 million tires for half-price replacement.

Complaints pertaining to the recall should be sent to either:

Mr. J.B. Scarcliff, Consumer Affairs Department, The Firestone Tire & Rubber Co., 1200 Firestone Parkway, Akron, Ohio 44317, or,

The National Highway Traffic Safety Administration, 400 7th St., S.W., Washington, D.C. 20590.